diff --git "a/29FAT4oBgHgl3EQfDxyZ/content/tmp_files/load_file.txt" "b/29FAT4oBgHgl3EQfDxyZ/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/29FAT4oBgHgl3EQfDxyZ/content/tmp_files/load_file.txt" @@ -0,0 +1,1130 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf,len=1129 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='08418v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='CT] 20 Jan 2023 Measurings of Hopf algebroids and morphisms in cyclic (co)homology theories Abhishek Banerjee * Surjeet Kour † Abstract In this paper, we consider measurings between Hopf algebroids and show that they induce morphisms on cyclic homology and cyclic cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We also consider comodule measurings between SAYD modules over Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' These measur- ings induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Finally, we obtain morphisms on cyclic homology induced by measurings of cyclic comp modules over operads with multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' MSC(2020) Subject Classification: 16T15, 16E40, 18D50 Keywords: Hopf algebroids, cyclic (co)homology, SAYD modules, comp modules 1 Introduction Let k be a field, C be a k-coalgebra and A, B be k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In [20], Sweedler introduced the notion of a coalgebra measuring as a kind of generalized morphism between algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' More precisely, a C-measuring from A to B consists of a k-linear map φ : C −→ Vectk(A, B) satisfying φ(c)(aa′) = � φ(c(1)(a)φ(c(2))(a′) φ(c)(1) = ǫ(c)1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) for any a, a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Here, ∆(c) = � c(1) ⊗ c(2) denotes the coproduct on C and ǫ : C −→ k denotes the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Since then, the notion of a measuring has been widely studied in the literature by several authors (see, for instance, [1], [3], [4], [9], [10], [11], [12],[22], [23], [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In [2], we studied how coalgebra measurings induce morphisms between Hochschild homology groups of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The purpose of this paper is to take this idea one step further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Our aim is to consider cocommutative coalgebra morphisms between Hopf algebroids and show that they induce morphisms in cyclic homology and cyclic cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We begin by showing that there are universal measurings which give an enrichment of the category HAlgk of Hopf algebroids over the category of cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' If U is a Hopf algebroid, the cyclic module C•(U) defining its cyclic homology groups as well as the cocyclic module C•(U) defining its cyclic cohomology groups are defined in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We show that a cocommutative coalgebra measuring between two Hopf algebroids induces morphisms on their corresponding cyclic homology and cyclic cohomology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' If a Hopf algebroid is commutative, we know from [15] that there is a shuffle product on its Hochschild homology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We show that a measuring between commutative Hopf algebroids induces an algebra measuring between the corresponding Hochschild homology rings with respct to this shuffle product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' This also gives us an enrichment of commutative Hopf algebroids over cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Thereafter, we consider comodule measurings of SAYD modules over Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Accordingly, we obtain an enrichment of the “global category” of SAYD modules (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9) over the “global category” of comodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The cyclic homology Department of Mathematics, Indian Institute of Science, Bangalore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Email: abhishekbanerjee1313@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='com †Department of Mathematics, Indian Institute of Technology, Delhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Email: koursurjeet@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='com 1 and the cyclic cohomology of a Hopf algebroid with coefficients in an SAYD module was defined in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We show that a comodule measuring induces morphisms between cyclic (co)homology with SAYD coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In the final part of the paper, we work with pairs of the form (O, M), where M is a cyclic unital comp module over a non-Σ operad O with multiplication in the sense of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The cyclic homology groups of such a comp module were also defined in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We consider comodule measurings between such pairs and show that they induce morphisms in cyclic homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 2 Measurings of Hopf algebroids Throughout, k is a field and let Vectk be the category of k-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let A be a unital k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In order to define left and right bialgebroids, as well as Hopf algebroids in later sections, we will frequently need both the algebra A and its opposite algebra Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For this, we will often write the algebra A as AL, while Aop will often be written as AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' An (s, t)-ring over A consists of a unital k-algebra U along with two k-algebra morphisms s : A −→ U and t : Aop −→ U whose images commute in U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=', s(a1)t(a2) = t(a2)s(a1) for any a1, a2 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The morphisms s and t are often referred to as source and target maps respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' These morphisms introduce an (A, A)-bimodule structure on U given by left multiplication a1 · h · a2 := s(a1)t(a2)h a1, a2 ∈ A, h ∈ H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) The left and right A-module structures on U in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) allow us to consider the tensor product U ⊗A U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The following subspace of U ⊗A U is known as the Takeuchi product U ×A U := {� ui ⊗A u′ i ∈ U ⊗A U | � uit(a) ⊗A u′ i = � ui ⊗A u′ is(a), ∀ a ∈ A} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2) It is well known (see, for instance, [13, § 2]) that the Takeuchi product U ×A U is a unital subalgebra of U ⊗A U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From now onwards, we also fix a unital k-algebra U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The multiplication on U will be denoted by µU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Since the category of (A, A)-bimodules is monoidal, we can consider coalgebra objects in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now recall the notion of a left Hopf algebroid (see, for instance, [5], [13], [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For several closely related notions, see [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A left bialgebroid UL := (U, AL, sL, tL, ∆L, ǫL) over k consists of the following data: (1) A unital k-algebra AL (2) A unital k-algebra U which carries the structure of an (sL, tL) ring over AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (3) A coalgebra object (U, ∆L : U −→ U ⊗AL U, ǫL : U −→ AL) in the category of (AL, AL)-bimodules satisfying the following conditions: (i) ∆L : U −→ U ⊗AL U factors through U ×A U ⊆ U ⊗AL U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (ii) ǫL(usL(ǫL(u′))) = ǫL(uu′) = ǫL(utL(ǫL(u′))) for all u, u′ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A morphism (F, f) : (U, AL, sL, tL, ∆L, ǫL) = UL −→ U′ L = (U′, A′ L, s′ L, t′ L, ∆′ L, ǫ′ L) of left bialgebroids consists of a pair of k-algebra morphisms F : U −→ U′ and f : AL −→ A′ L such that F ◦ sL = s′ L ◦ f F ◦ tL = t′ L ◦ f ∆′ L ◦ F = (F ⊗φ F) ◦ ∆L f ◦ ǫL = ǫ′ L ◦ F (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3) We will denote the category of left bialgebroids over k by LBialgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' If UL = (H, AL, sL, tL, ∆L, ǫL) is a left bialgebroid, we employ standard Sweedler notation to write ∆L(u) = � u(1) ⊗ u(2) for any u ∈ H and suppress the summation sign throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now recall the notion of Hopf algebroid from [5, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A Hopf algebroid U = (UL, S ) over k consists of the following data: (1) A left bialgebroid UL = (U, AL, sL, tL, ∆L, ǫL) over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (2) An involutive anti-automorphism S : U −→ U of the k-algebra U which satisfies S ◦ tL = sL as well as S (u(1))(1)u(2) ⊗ S (u(1))(2) = 1H ⊗ S (u) S (u2)1 ⊗ S (u(2))(2)u(1) = S (u) ⊗ 1U (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4) 2 as elements of U ⊗AL U, for all u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A morphism (F, f) : U = (UL, S ) −→ (U′ L, S ′) = U′ of Hopf algebroids is a morphism in LBialgk that also satisfies S ′◦F = f ◦S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We will denote the category of Hopf bialgebroids over k by HAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We remark here that in this paper we will always assume the antipode on a Hopf algebroid U = (UL, S ) is involutive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=', S 2 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' However, this condition is not part of the original definition due to B¨ohm and Szlach´anyi in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Further, it is shown in [5, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2] that a Hopf algebroid (UL, S ) is equivalent to a datum consisting of a left bialgebroid and a right bialgebroid connected by an antipode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now recall the classical notion of a coalgebra measuring due to Sweedler [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let R, R′ be k-algebras and C be a k- coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, a C-measuring from R to R′ consists of a morphism ψ : C −→ Vectk(R, R′) such that ψ(x)(ab) = � ψ(x(1))(a)ψ(x(2)) ψ(x)(1R) = ǫC(x)1R′ ∀ a, b ∈ R (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) where the coproduct ∆C : C −→ C ⊗ C is given by ∆C(x) = � x(1) ⊗ x(2) for any x ∈ C and ǫC : C −→ k is the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The measuring as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) is said to be cocommutative if the coalgebra C is cocommutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In this paper, we will only consider cocommutative measurings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By abuse of notation, if ψ : C −→ Vectk(R, R′) is a coalgebra measuring, we will often write the morphism ψ(x) ∈ Vectk(R, R′) simply as c : R −→ R′ for any x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We are now ready to introduce the notion of measuring between Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′ L, t′ L, ∆′ L, ǫ′ L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative k-coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A C-measuring (Ψ, ψ) from U to U′ consists of a pair of measurings Ψ : C −→ Vectk(U, U′) ψ : C −→ Vectk(AL, A′ L) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6) such that the following diagrams commute for any x ∈ C AL sL −−−−−−→ U c \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c A′ L s′ L −−−−−−→ U′ AL tL −−−−−−→ U c \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c A′ L t′ L −−−−−−→ U′ U S −−−−−−→ U c \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c U′ S ′ −−−−−−→ U′ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7) U ǫL −−−−−−→ AL c \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c U′ ǫ′ L −−−−−−→ A′ L U ∆L −−−−−−→ U ⊗AL U c \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c U′ ∆′ L −−−−−−→ U′ ⊗A′ L U′ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8) where the arrow c : U ⊗AL U −→ U′ ⊗A′ L U′ is defined by setting c(u1 ⊗ u2) := x(1)(h) ⊗ x(2)(u2) for u1 ⊗ u2 ∈ U ⊗AL U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Before proceeding further, we need to verify the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For any x ∈ C, the morphism c : H ⊗AL U −→ U′ ⊗A′ L U′ defined by setting c(u1 ⊗ u2) := x(1)(h) ⊗ x(2)(u2) for u1 ⊗ u2 ∈ U ⊗AL U is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We consider u1, u2 ∈ uL and a ∈ AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and applying the conditions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8), we see that c((u1 · a) ⊗ u2) = c(tL(a)u1 ⊗ u2) = x(1)(tL(a)u1) ⊗ x(2)(u2) = x(1)(tL(a))x(2)(u1) ⊗ x(3)(u2) = t′ L(x(1)(a))x(2)(u1) ⊗ x(3)(u2) = t′ L(x(2)(a))x(1)(u1) ⊗ x(3)(u2) (because C is cocommutative) = x(1)(u1) · x(2)(a) ⊗ x(3)(u2) = x(1)(u1) ⊗ x(2)(a) · x(3)(u2) = x(1)(u1) ⊗ s′ L(x(2)(a))x(3)(u2) = x(1)(u1) ⊗ x(2)(sL(a))x(3)(u2) = x(1)(u1) ⊗ x(2)(sL(a)u2) = x(1)(u1) ⊗ x(2)(a · u2) = c(u1 ⊗ (a · u2)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9) □ 3 If U = (UL, S ) and U′ = (U′ L, S ′) are Hopf algebroids over k, we now consider the subspace V(U, U′) ⊆ Vectk(U, U′) × Vectk(AL, A′ L) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='10) given by setting V(U, U′) := {(F, f) | FsL = s′ L f, FtL = t′ L f, FS = S ′F and fǫL = ǫ′ LF } (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) We note that a measuring from U to U′ by means of a cocommutative coalgebra C has an underlying morphism (Ψ, ψ) : C −→ V(U, U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let Coalgk denote the category of k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We know that the forgetful functor Coalgk −→ Vectk has a right adjoint C : Vectk −→ Coalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In other words, we have natural isomorphisms Vectk(C, V) � Coalgk(C, C(V)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) for any k-coalgebra C and any k-vector space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U = (UL, S ) and U′ = (U′ L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, there exists a cocommutative coalgebra Mc(U, U′) and a measuring (Φ, φ) from U to U′ satisfying the following universal property: given any measuring (Ψ, ψ) : C −→ V(U, U′) with a cocommutative coalgebra C, there exists a unique morphism ξ : C −→ Mc(U, U′) of coalgebras making the following diagram commutative Mc(U, U′) (Φ,φ) � V(U, U′) C ξ �■■■■■■■■■■ (Ψ,ψ) �✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We set V := V(U, U′) and consider the canonical morphism π(V) : C(V) −→ V ⊆ Vectk(uL, u′ L) × Vectk(AL, A′ L) from the cofree coalgebra C(V) induced by the adjunction in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now set Mc(U, U′) := � D, where the sum is taken over all cocommutative subcoalgebras of C(V) such that the restriction π(V)|D : D −→ V = V(U, U′) is a measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It is clear that this sum is still a cocommutative coalgebra, and that the restriction (Φ, φ) := π(V)|Mc(U,U′) gives a measuring from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In general, if (Ψ, ψ) : C −→ V = V(U, U′) is a cocommutative measuring, the adjunction in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) shows that it factors through ξ : C −→ C(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, ξ(C) ⊆ C(V) is a cocommutative coalgebra such that the restriction π(V)|ξ(C) is a measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By definition, it follows that ξ(C) ⊆ Mc(U, U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) it is clear that given Hopf algebroids U = (UL, S ), U′ = (U′ L, S ′) and U′′ = (U′′ L, S ′′), the composition of morphisms induces a canonical map V(U, U′) ⊗ V(U′, U′′) −→ V(U, U′′) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='14) We denote by CoCoalgk the category of cocommutative coalgebras over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We know that this category is symmetric monoidal and our objective is to show that the category HAlgk of Hopf algebroids is enriched over CoCoalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For this we need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U = (UL, S ), U′ = (U′ L, S ′) and U′′ = (U′′ L, S ′′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Suppose that we have a measuring (Ψ, ψ) : C −→ V(U, U′) from U to U′ and a measuring (Ψ′, ψ′) : C′ −→ V(U′, U′′) from U′ to U′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, the following (Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ (Ψ,ψ)⊗(Ψ′,ψ′) −−−−−−−−−−→ V(U, U′) ⊗ V(U′, U′′) −→ V(U, U′′) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='15) determines a measuring from U to U′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It is easy to verify that the compositions C ⊗ C′ Ψ⊗Ψ′ −−−−→ Vectk(U, U′) ⊗ Vectk(U′, U′′) −−−−−−→ Vectk(U, U′′) C ⊗ C′ ψ⊗ψ′ −−−−→ Vectk(AL, A′ L) ⊗ Vectk(A′ L, A′′ L) −−−−−−→ Vectk(AL, A′′ L) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='16) 4 give coalgebra measurings from U to U′′ and from AL to A′′ L respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For c ⊗ c′ ∈ C ⊗ C′ and u ∈ U, we also see that ∆′′ L((c ⊗ c′)(u)) = ∆′′ L(c′(c(u))) = c′ (1)(c(u)(1)) ⊗ c′ (2)(c(u)(2)) = c′ (1)(x(1)(u(1))) ⊗ c′ (2)(x(2)(u(2))) = (c′ ⊗ c)(1)(u(1)) ⊗ (c′ ⊗ c)(2)(u(2)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='17) It is also clear that the morphism in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='15) satisfies all the other conditions in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The category HAlgk of Hopf algebroids is enriched over the category CoCoalgk of cocommutative k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Given Hopf algebroids U = (UL, S ) and U′ = (U′ L, S ′), we consider the “hom object” Mc(U, U′) which lies in CoCoalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The composition of these hom objects is obtained as follows: if U, U′ and U′′ are Hopf algebroids, we obtain as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6 a measuring Mc(U, U′) ⊗ Mc(U′, U′′) −→ V(U, U′′) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='18) Applying the universal property in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5, we now have a morphism of coalgebras Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The unit object in CoCoalgk is k treated as a coalgebra over itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, we have a unit map k −��� V(U, U) ⊆ Vectk(U, U) × Vectk(AL, AL) t �→ (t · iduL, t · idAL) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='19) which induces a morphism k −→ Mc(U, U) of cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Together with the composition of hom objects in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='18), we see that HAlgk is enriched over CoCoalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ From now onwards, we will denote by HALGk the category of Hopf algebroids enriched over the symmetric monoidal category CoCoalgk of cocommutative k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 3 Morphisms on cyclic (co)homology and Hopf-Galois maps Let U = (UL, S ) = (U, AL, sL, tL, ∆L, ǫL) be a Hopf algebroid over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now recall from [13, § 2] the cocyclic module C•(U) that computes the cyclic cohomology of the Hopf algebroid U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For n ≥ 1, we put Cn(U) := U ⊗AL ⊗ · · · ⊗AL U �������������������������������������� n-times (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) and set C0(U) := AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For n ≥ 1, the face maps δi : Cn(U) −→ Cn+1(U) are defined by δi(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 1 ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un if i = 0 u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ ∆Lui ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un if 1 ≤ i ≤ n u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ un ⊗ 1 if i = n + 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2) For n = 0, there are two maps δ0 := tL : C0(U) = AL −→ C1(U) = U and δ1 := sL : C0(U) = AL −→ C1(U) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The degeneracy maps σi : Cn(U) −→ Cn−1(U) are given by σi(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) := u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ ǫL(ui+1) · ui+2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un 0 ≤ i ≤ n − 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3) The cyclic operator τn : Cn(U) −→ Cn(U) is defined by setting τn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) := (S (u1)(1) · u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ (S (u1)(n−1) · un) ⊗ S (u1)(n) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4) Since we have assumed that the antipode S is involutive, it follows from [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1] that C•(U) is indeed a cocyclic module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We will denote by HC•(U) the cyclic cohomology groups of the Hopf algebroid U by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The Hochschild cohomology groups of the Hopf algebroid U will then be denoted by HH•(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 5 Let U, U′ be Hopf algebroids and let (Ψ, ψ) : C −→ V(U, U′) be a measuring from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For each x ∈ C, we now define a family of morphisms Ψ n(x) : Cn(U) −→ Cn(U′) Ψ n(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) := x(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(un) ∀ n ≥ 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) We now prove the first main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For each x ∈ C, the family {Ψ n(x) : Cn(U) −→ Cn(U′)}n≥0 gives a morphism of cyclic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In particular, we have induced morphisms Ψ hoc(x) : HH•(U) −→ HH•(U) Ψ cy(x) : HC•(U) −→ HC•(U) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6) on Hochschild and cyclic cohomologies for each x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For each x ∈ C, we start by showing that Ψ n+1(x) ◦ δi = δ′ i ◦ Ψ n(x) : Cn(U) −→ Cn+1(H ′), where δi and δ′ i are the face maps on the respective cocyclic modules C•(U) and C•(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' If i = 0 or i = n + 1, this is immediately clear from the definition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2) and the action in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For 1 ≤ i ≤ n, we see that Ψ n+1(x) ◦ δi(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) = Ψ n+1(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ ∆Lui ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(i)(ui (1)) ⊗ x(i+1)(ui (2)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ x(n+1)(un) = x(1)(u1) ⊗ ∆L(x(i)(ui)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(un) = δ′ i ◦ Ψ n(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) Next, we verify that Ψ n−1(x) ◦ σi = σ′ i ◦ Ψ n(x), where σi and σ′ i are the degeneracies on the respective cocyclic modules C•(U) and C•(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Ψ n−1(x) ◦ σi(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) = Ψ n−1(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ ǫL(ui+1) · ui+2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(i+1)(ǫL(ui+1) · ui+2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n−1)(un) = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(i+1)(ǫL(ui+1)) · x(i+2)(ui+2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(un) = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ ǫL(x(i+1)(ui+1)) · x(i+2)(ui+2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(un) = σ′ i ◦ Ψ n(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) Finally, we show that Ψ n(x)◦τn = τ′ n ◦Ψ n(x), where τn and τ′ n are the cyclic operators on the respective cocyclic modules C•(U) and C•(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Ψ n(x) ◦ τn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) = Ψ n(x)((S (u1)(1) · u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ (S (u1)(n−1) · un) ⊗ S (u1)(n)) = x(1)((S (u1)(1) · u2)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ x(n−1)((S (u1)(n−1) · un) ⊗ x(n)(S (u1)(n)) = x(1)(S (u1)(1)) · x(2)(u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ x(2n−3)(S (u1)(n−1)) · x(2n−2)(un) ⊗ x(2n−1)(S (u1)(n)) = x(1)(S (u1)(1)) · x(n+1)(u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ x(n−1)(S (u1)(n−1)) · x(2n−1)(un) ⊗ x(n)(S (u1)(n)) = (x(1)(S (u1)))(1) · x(2)(u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ (x(1)(S (u1)))(n−1) · x(n)(un) ⊗ (x(1)(S (u1)))(n) = (S (x(1)(u1)))(1) · x(2)(u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ (S (x(1)(u1)))(n−1) · x(n)(un) ⊗ (S (x(1)(u1)))(n) = τ′ n ◦ Ψ n(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) □ We continue with a Hopf algebroid U = (UL, S ) = (U, AL, sL, tL, ∆L, ǫL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' As mentioned in Section 2, we set AR := Aop L = Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Following [5, § 4], we also set sR := tL tR := S ◦ tL = sL (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7) Then, U becomes an (AR, AR)-bimodule by right multiplication as follows a1 · h · a2 := hsR(a2)tR(a1) = htL(a2)sL(a1) h ∈ H, a1, a2 ∈ AR (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8) We now consider S rl : H ⊗AR H −→ H ⊗AL H u1 ⊗ u2 �→ S (u2) ⊗ S (u1) S lr := S −1 rl : H ⊗AL H −→ H ⊗AR H u1 ⊗ u2 �→ S (u2) ⊗ S (u1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9) 6 as well as ∆R := S lr ◦ ∆L ◦ S : U −→ U ⊗AR U ǫR := ǫL ◦ S : U −→ AR (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='10) We know from [5, § 4] that the datum UR := (U, AR, sR, tR, ∆R, ǫR) defines a right bialgebroid over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now adopt the Sweedler notation ∆R(u) := u[1] ⊗ u[2] for any u ∈ U in order to distinguish it from the left coproduct ∆L(u) = u(1) ⊗ u(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' More explicitly, we have ∆R(u) = u[1] ⊗ u[2] = S (S (u)(2)) ⊗ S (S (u)(1)) ∀ u ∈ U (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) Now let U, U′ be Hopf algebroids and consider (F, f) ∈ V(U, U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From the conditions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) and the definitions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='10), we already have FsR = s′ R f FtR = t′ R f FS = S ′F fǫR = ǫ′ RF (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) We now need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative coalgebra and (Ψ, ψ) : C −→ V(U, U′) a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, for each x ∈ C, there is a well defined morphism x : U −→ U ⊗AR U u1 ⊗ u2 �→ x(1)(u1) ⊗ x(2)(u2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='13) which fits into the following commutative diagram U ∆R −−−−−−→ U ⊗AR U x \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�x U′ ∆′ R −−−−−−→ U′ ⊗A′ R U′ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We consider u1, u2 ∈ uL and a ∈ AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and applying the conditions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12), we see that c((u1 · a) ⊗ u2) = c(u1sR(a) ⊗ u2) = x(1)(u1sR(a)) ⊗ x(2)(u2) = x(1)(u1)x(2)(sR(a)) ⊗ x(3)(u2) = x(1)(u1)s′ R(x(2)(a)) ⊗ x(3)(u2) = x(1)(u1) · x(2)(a) ⊗ x(3)(u2) = x(1)(u1) ⊗ x(2)(a) · x(3)(u2) = x(1)(u1) ⊗ x(3)(u2)t′ R(x(2)(a)) = x(1)(u1) ⊗ x(3)(u2)x(2)(tR(a)) = x(1)(u1) ⊗ x(2)(u2)x(3)(tR(a)) (as C is cocommutative) = x(1)(u1) ⊗ x(2)(u2tR(a)) = x(1)(u1) ⊗ x(2)(a · u2) = c(u1 ⊗ (a · u2)) It follows that the morphism in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='13) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It remains to verify the condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For u ∈ U and x ∈ C, we have c(∆R(u)) = c(S (S (u)(2)) ⊗ S (S (u)(1))) = x(1)(S (S (u)(2))) ⊗ x(2)(S (S (u)(1))) = S ′(x(1)(S (u)(2))) ⊗ S ′(x(2)(S (u)(1))) = S ′(x(2)(S (u)(2))) ⊗ S ′(x(1)(S (u)(1))) (as C is cocommutative) = S ′(c(S (u))(2)) ⊗ S ′(c(S (u))(1)) = S ′(S ′(c(u))(2)) ⊗ S ′(S ′(c(u))(1)) = ∆′ R(c(u)) □ We now recall from [13, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1] the cyclic module C•(U) defining the cyclic homology of a Hopf algebroid U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For n ≥ 0, we set Cn(U) := U ⊗AR ⊗ · · · ⊗AR U �������������������������������������� n-times (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='15) 7 and C0(U) := AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The face maps di : Cn(U) −→ Cn−1(U) are defined by setting di(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 ǫR(u1)u2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un if i = 0 u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ uiui+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un if i ≤ i ≤ n − 1 u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un−1ǫR(S (un)) if i = n (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='16) The degeneracies si : Cn(U) −→ Cn+1(U) are defined as si(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) := � 1 ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un if i = 0 u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ ui ⊗ 1 ⊗ ui+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='. ⊗ un if 1 ≤ i ≤ n (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='17) The cyclic operators tn : Cn(U) −→ Cn(U) are given by tn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) := S (u1 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='un−1 (2) un) ⊗ u1 (1) ⊗ u2 (1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un−1 (1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='18) The Hochschild homology groups of the Hopf algebroid U will then be denoted by HH•(U) and the cyclic homology groups by HC•(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We will now prove the homological counterpart for Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For each x ∈ C, the family Ψn(x) : Cn(U) −→ Cn(U′) u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un �→ x(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(un) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='19) for n ≥ 0 gives a morphism of cyclic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In particular, we have induced morphisms Ψhoc (x) : HH•(U) −→ HH•(U′) Ψcy (x) : HC•(U) −→ HC•(U′) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='20) on Hochschild and cyclic homologies for each x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Using the properties in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) and the fact that Ψ : C −→ Vectk(U, U′) is a measuring, it may easily be verified that the maps Ψ•(x) commute with the respective face maps and degeneracy maps on the cyclic modules C•(U) and C•(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Moreover, if tn and t′ n are the respective cyclic operators on C•(U) and C•(U′), we have for each x ∈ C c(tn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un)) = c(S (u1 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='un−1 (2) un) ⊗ u1 (1) ⊗ u2 (1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un−1 (1) ) = x(1)(S (u1 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='un−1 (2) un)) ⊗ x(2)(u1 (1)) ⊗ x(3)(u2 (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(un−1 (1) ) = S ′(x(1)(u1 (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='x(n−1)(un−1 (2) )x(n)(un)) ⊗ x(n+1)(u1 (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(2n−1)(un−1 (1) ) = S ′(x(2)(u1 (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='x(2n−2)(un−1 (2) )x(2n−1)(un)) ⊗ x(1)(u1 (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(2n−3)(un−1 (1) ) = S ′(x(1)(u1)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='x(n−1)(un−1)(2)x(n)(un)) ⊗ x(1)(u1)(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n−1)(un−1)(1) = t′ n(x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(un)) □ Our final aim in this section is to show that the morphisms induced by a measuring of Hopf algebroids are well behaved with respect to cyclic duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' More precisely, we know from [13, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3] that there are Hopf-Galois maps ξn(U) : Cn(U) � −→ Cn(U) u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un �→ u1 (1) ⊗ u1 (2)u2 (1) ⊗ u1 (3)u2 (2)u3 (1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ u1 (n)u2 (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='.un−1 (2) un (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='21) inducing isomorphisms between C•(U) and C•(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then for each x ∈ C, the following diagram commutes Cn(U) ξn(U) −−−−−−→ Cn(U) Ψn(x) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�Ψ n(x) Cn(U) ξn(U) −−−−−−→ Cn(U) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='22) 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We put N := n(n + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and that C is cocommutative we have c(ξn(U)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un)) = c(u1 (1) ⊗ u1 (2)u2 (1) ⊗ u1 (3)u2 (2)u3 (1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ u1 (n)u2 (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='.un−1 (2) un) = x(1)(u1 (1)) ⊗ x(2)(u1 (2))x(3)(u2 (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(N+1−n)(u1 (n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='.x(N−1)(un−1 (2) )x(N)(un) = x(1)(u1 (1)) ⊗ x(2)(u1 (2))x(n+1)(u2 (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(u1 (n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='.x(N−1)(un−1 (2) )x(N)(un) = ξn(U′)(x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(n)(un)) This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ 4 Shuffle products and the enrichment of the category of commutative Hopf alge- broids We recall from Section 2 the category HALGk of Hopf algebroids over k, enriched over the symmetric monoidal category of CoCoalgk of cocommutative k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By a commutative Hopf algebroid, we will mean a Hopf algebroid U = (U, AL, sL, tL, ∆L, ǫL) such that H and AL = A = AR are commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let cHALGk denote the full subcategory of HALGk consisting of commutative Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, cHALGk is also en- riched over CoCoalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In this section, we will obtain a second enrichment of commutative Hopf algebroids in cocommutative coalgebras, by using the shuffle product in Hochschild homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We know from [17, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2] that the Hochschild homology of a commutative algebra is equipped with a shuffle product structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For a commutative Hopf algebroid U = (U, AL, sL, tL, ∆L, ǫL), we now recall from [15, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1] the (p, q)-shuffle product shpq(U) : Cp(U) ⊗ Cq(U) −→ Cp+q(U) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) which is given by the formula (for p, q ≥ 1) shpq(U)((u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) ⊗ (up+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up+q)) := � σ∈S h(p,q) sgn(σ)(uσ−1(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ uσ−1(p+q)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2) Here S h(p, q) is the set of (p, q)-shuffles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=', S h(p, q) := {σ ∈ S p+q | σ(1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' < σ(p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' σ(p + 1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' < σ(p + q)} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3) For p = q = 0, the shuffle product is given by setting sh00(U) to be the multiplication on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Further, one has (see [15, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1]) shp0(U) : Cp(U) ⊗ C0(U) −→ Cp(U) (u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) ⊗ a �→ (tL(a)u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) sh0q(U) : C0(U) ⊗ Cq(U) −→ Cq(U) a ⊗ (u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) �→ (u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ uqtL(a)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4) for p ≥ 1 and q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' There is now an induced product structure shpq(U) : HHp(U) ⊗ HHq(U) −→ HHp+q(U) which makes the the Hochschild homology HH•(U) := � n≥0 HHp(U) of a commutative Hopf algebroid U into a graded algebra (see [15, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1]) that we denote by (HH•(U), sh(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U, U′ be commutative Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, the induced K-linear map Ψhoc : C −→ HomK(HH•(U), HH•(U′)) x �→ (Ψhoc (x) : HH•(U) −→ HH•(U′)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) gives a measuring of algebras from (HH•(U), sh(U)) to (HH•(U′), sh(U′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The unit in (HH•(U), sh(U)) is given by the class of the unit 1A ∈ A = C0(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Since ψ : C −→ HomK(A, A′) gives in particular a measuring from A to A′, we have Ψhoc (x)(1A) = 1A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now note that for any x ∈ C and p, q ≥ 1, we have Ψp+q(x)(shpq(U)((u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) ⊗ (up+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up+q))) = Ψp+q(x) � � σ∈S h(p,q) sgn(σ)(uσ−1(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ uσ−1(p+q)) � = � σ∈S h(p,q) sgn(σ)(x(1)(uσ−1(1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(p+q)(uσ−1(p+q))) = � σ∈S h(p,q) sgn(σ)(xσ−1(1)(uσ−1(1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ xσ−1(p+q)(uσ−1(p+q))) (because C is cocommutative) = shpq(U)((x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(p)(up)) ⊗ (x(p+1)(up+1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(p+q)(up+q))) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6) For p ≥ 1, we have Ψp(x)(shp0(U)((u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) ⊗ a) = Ψp(x)((tL(a)u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up)) = (x(1)(tL(a)u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(p)(up)) = (x(1)(tL(a))x(2)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(p+1)(up)) = (tL(x(p+1)(a)))x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(p)(up)) = shp0(U)((x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ x(p)(up)) ⊗ x(p+1)(a)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7) We can similarly verify the case for sh0q with q ≥ 1 and for sh00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ Our next objective is to use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1 to obtain an enrichment of commutative Hopf algebroids over the category of cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For that we recall the following fact: if R, R′ are k-algebras, the category of coalgebra measurings from R to R′ contains a final object ϕ(R, R′) : M(R, R′) −→ Vectk(R, R′) (see Sweedler [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, M(R, R′) is known as the universal measuring coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We let Mc(R, R′) be the cocommutative part of the coalgebra M(R, R′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, the restriction ϕc(R, R′) : Mc(R, R′) ֒→ M(R, R′) −→ Vectk(R, R′) becomes the final object in the category of cocommutative coalgebra measurings from R to R′ (see [9, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4], [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Further, the objects Mc(R, R′) give an enrichment of k-algebras over cocommutative k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now define the enriched category � cHALGk whose objects are commutative Hopf algebroids over k and whose hom-objects are defined by setting � cHALGk(U, U′) := Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) ∈ CoCoalgk (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8) for commutative Hopf algebroids U, U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Since each (HH•(U), sh(U)) is an algebra, we also have a canonical morphism k −→ Mc((HH•(U), sh(U)), (HH•(U), sh(U))) of cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U, U′ be commutative Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, there is a canonical morphism of cocommutative coalgebras τ(U, U′) : Mc(U, U′) −→ Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By definition, (Φ, φ) : Mc(U, U′) −→ V(U, U′) is a cocommutative measuring from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1, this induces a measuring of algebras from (HH•(U), sh(U)) to (HH•(U′), sh(U′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By the universal property of the universal cocommutative measuring coalgebra Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))), we now obtain an induced morphism τ(U, U′) as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' There is a CoCoalgk enriched functor cHALGk −→ � cHALGk which is identity on objects and whose mapping on hom-objects is given by τ(U, U′) : cHALGk(U, U′) = Mc(U, U′) −→ Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) = � cHALGk(U, U′) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='10) for commutative Hopf algebroids U, U′ over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U, U′, U′′ be commutative Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We show that the following diagram commutes Mc(U, U′) ⊗ Mc(U′, U′′) −−−−−−→ Mc(U, U′′) τ(U,U′)⊗τ(U′,U′′) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�τ(U,U′′) Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′)) −−−−−−→ Mc(HH•(U), HH•(U′′)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) The top horizontal composition ◦ : Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) is obtained from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7, while the bottom horizontal composition ◦ : Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′)) −→ Mc(HH•(U), HH•(U′′)) is obtained from the enrichment of algebras in cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7, we note that all the maps in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) are morphisms of cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It follows from the property of the universal cocommutative measuring coalgebra Mc(HH•(U), HH•(U′′)) that in order to show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) commutes, it suffices to verify that the following two compositions are equal Mc(U, U′) ⊗ Mc(U′, U′′) τ(U,U′)⊗τ(U′,U′′) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′)) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� Mc(HH•(U), HH•(U′′)) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�ϕc(HH•(U),HH•(U′′)) Vectk(HH•(U), HH•(U′′)) Mc(U, U′) ⊗ Mc(U′, U′′) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�◦ Mc(U, U′′) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�τ(U,U′′) Mc(HH•(U), HH•(U′′)) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�ϕc(HH•(U),HH•(U′′)) Vectk(HH•(U), HH•(U′′)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) For the sake of convenience, we denote the left vertical composition in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) by ψ1 and the right vertical composition by ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now consider x ∈ Mc(U, U′), y ∈ Mc(U′, U′′) and (u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) ∈ Cp(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We see that ψ2(x ⊗ y)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) = (y ◦ x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) = (y ◦ x)(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ (y ◦ x)(p)(up) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='13) Since ◦ : Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) is a morphism of coalgebras, we note that (y ◦ x)(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ (y ◦ x)(p) = (y(1) ◦ x(1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ (y(p) ◦ x(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='13), we see that the right vertical composition in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) may be described explicitly as ψ2(x ⊗ y)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) = (y(1) ◦ x(1))(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ (y(p) ◦ x(p))(up) = y(1)(x(1)(u1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ y(p)(x(p)(up)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='14) On the other hand, we note that the following diagram is commutative Mc(U, U′) ⊗ Mc(U′, U′′) (τ(U,U′)⊗τ(U′,U′′)) −−−−−−−−−−−−−−−−→ Mc(HH•(U), HH•(U′′)) (ϕc(HH•(U),HH•(U′))◦τ(U,U′))⊗ \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�(ϕc(HH•(U′),HH•(U′′))◦τ(U′,U′′)) ϕc(HH•(U),HH•(U′′)) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� Vectk(HH•(U), HH•(U′)) ⊗ Vectk(HH•(U′), HH•(U′′)) −−−−−−→ Vectk(HH•(U), HH•(U′′)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='15) From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='15), it follows that the left vertical composition in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) may be described explicitly as ψ1(x ⊗ y)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up) = y(x(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ up)) = y(1)(x(1)(u1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ y(p)(x(p)(up)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='16) From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='16), we see that ψ1 = ψ2 and hence the diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Similarly by considering the coalgebra k and using the fact that the p-th iterated coproduct ∆p(1) = 1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ 1(p-times), we see that the following compositions are equal k −→ Mc(HH•(U), HH•(U)) ϕc(HH•(U),HH•(U)) −−−−−−−−−−−−−−−→ Vectk(HH•(U), HH•(U)) k −→ Mc(U, U) τ(U,U) −−−−−→ Mc(HH•(U), HH•(U)) ϕc(HH•(U),HH•(U)) −−−−−−−−−−−−−−−→ Vectk(HH•(U), HH•(U)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='17) 11 It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='17) that the following diagram commutes k � �● Mc(HH•(U), HH•(U)) Mc(U, U) τ(U,U) �❧ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='18) This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ 5 Comodule measurings for SAYD modules Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From now onwards, we set Ae := A ⊗k Aop and define ηL : Ae = A ⊗k Aop sL⊗tL −−−−→ U ⊗ U −→ U (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) where the second arrow in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) is the multilplication on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Following [14, § 2], we note that there are now four commuting actions of A on U which are denoted as follows a ⊲ u ⊳ b := sL(a)tL(b)u a ◮ u ◭ b := usL(b)tL(a) a, b ∈ A, u ∈ U (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2) By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1, we then have an A-coring ∆L : U −→ U⊳ ⊗A ⊲U ǫL : U −→ A (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3) The left action ◮ of A on U may be treated as a right action of Aop on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Similarly, the right action ⊳ of A on U may be treated as a left action by Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Accordingly, we may consider the tensor product ◮U ⊗Aop U⊳ := U ⊗k U/span{a ◮ u ⊗ v − u ⊗ v ⊳ a | u, v ∈ U, a ∈ A} (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4) There is now a Hopf-Galois map (see [5], [14], [19]) β(U) : ◮U ⊗Aop U⊳ −→ U⊳ ⊗A ⊲U u ⊗Aop v �→ u(1) ⊗A u(2)v (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) Since U is a Hopf algebroid, it follows (see [5, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2]) that the morphism β(U) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Accordingly, in the notation of [14], [19], we write u+ ⊗Aop u− := β(U)−1(u ⊗A 1) u ∈ U (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6) In this section, we will consider comodule measurings between stable anti-Yetter Drinfeld modules over Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For this, we first recall the notion of comodule measuring between ordinary modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let R, R′ be rings and let P, P′ be modules over R and R′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, a comodule measuring from P to P′ consists of a pair of maps (see [4], [12]) ψ : C −→ Vectk(R, R′) ω : D −→ Vectk(P, P′) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7) where C is a k-coalgebra, D is a right C-comodule, ψ : C −→ Vectk(R, R′) is a coalgebra measuring and ω(y)(pr) = y(pr) = y(0)(p)y(1)(r) = ω(y(0))(p)ψ(y(1))(r) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8) for y ∈ D, p ∈ P and r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For U = (U, AL, sL, tL, ∆L, ǫL, S ), we will now recall the notions of U-modules, U-comodules and stable anti-Yetter Drinfeld modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (see [14, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A right U-module P is a right module over the k-algebra U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Because of the ring homomorphism ηL : Ae −→ U, any right U-module P is also equipped with a right Ae-module structure (or (A, A)-bimodule structure) given by b ◮ p ◭ a = p(a ⊗ b) = pηL((a ⊗ 1)(1 ⊗ b)) = psL(a)tL(b) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9) for (a ⊗ b) ∈ Ae = A ⊗k Aop and p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 12 Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (see [6], [8], [14], [18]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A left U-comodule P is a left comodule over the A-coring (U, ∆L : U −→ U ⊗AL U, ǫL : U −→ AL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In particular, a left U-comodule P is equipped with a left A-module structure (a, p) �→ ap as well as a left A-module map ∆P : P −→ U⊳ ⊗A P p �→ p(−1) ⊗ p(0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='10) Following [14, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5], we note that any left U-comodule P also carries a right A-module structure given by setting pa := ǫL(p(−1)sL(a))p(0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) for p ∈ P, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' This makes any left U-comodule P into a right Ae = A ⊗k Aop-module by setting p(a ⊗ b) = bpa = bǫL(p(−1)sL(a))p(0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) for p ∈ P and (a ⊗ b) ∈ Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (see [14, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A stable anti-Yetter Drinfeld module (or SAYD module) P over U consists of the following (1) A right U-module structure on P denoted by (p, u) �→ pu for p ∈ P and u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (2) A left U-comodule structure on P given by ∆P : P −→ U⊳ ⊗A P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (3) The right Ae-module structure on P induced by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9) coincides with the right Ae-module structure on P as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12): psL(a)tL(b) = b ◮ p ◭ a = bǫL(p(−1)sL(a))p(0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='13) (4) For u ∈ U and p ∈ P, one has ∆P(pu) = u−p(−1)u+(1) ⊗A p(0)u+(2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='14) (5) Stability condition: for any p ∈ P, one has p(0)p(−1) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let R, R′ be k-algebras and let Re = R ⊗k Rop, R′e = R′ ⊗k R′op be their respective enveloping algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative k-coalgebra and let ψ : C −→ Vectk(R, R′) be a measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, ψe : C −→ Vectk(Re, R′e) ψe(c)(r ⊗ r′) = c(r1 ⊗ r2) = c(1)(r1) ⊗ c(2)(r2) = ψ(c(1))(r1) ⊗ ψ(c(2))(r2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='15) is a measuring of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='15), it is immediate that c(1 ⊗ 1) = ǫC(c)(1 ⊗ 1), where ǫC is the counit on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Since C is cocommutative, we have for (r1 ⊗ r2), (r3 ⊗ r4) ∈ Re c((r1 ⊗ r2)(r3 ⊗ r4)) = c(r1r3 ⊗ r4r2) = c(1)(r1r3) ⊗ c(2)(r4r2) = c(1)(r1)c(2)(r3) ⊗ c(3)(r4)c(4)(r2) = c(1)(r1)c(3)(r3) ⊗ c(4)(r4)c(2)(r2) = (c(1)(r1) ⊗ c(2)(r2))(c(3)(r3) ⊗ c(4)(r4)) = c(1)(r1 ⊗ r2)c(2)(r3 ⊗ r4) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='16) □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′ L, t′ L, ∆′ L, ǫ′ L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) be an SAYD-module over U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative k-coalgebra and D be a right C-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Suppose that we are given the following data Ψ : C −→ Vectk(U, U′) ψ : C −→ Vectk(A, A′) Ω : D −→ Vectk(P, P′) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='17) such that 13 (1) (Ψ, ψ) is a measuring of Hopf algebroids from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (2) (Ψ, Ω) is a comodule measuring from the right U-module P to the right U′ module P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, we have: (a) (ψe, Ω) is a comodule measuring from the right Ae-module P to the right A′e-module P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (b) For each d ∈ D, the following morphism is well-defined d : U⊳ ⊗A P −→ U′ ⊳ ⊗A′ P′ d(u ⊗A p) := Ψ(d(1))(u) ⊗A′ Ω(d(0))(p) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (a) Since C is cocommutative, we already know from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4 that ψe : C −→ Vectk(Ae, A′e) is a coalgebra measuring from Ae to A′e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now consider (a ⊗ b) ∈ Ae = A ⊗k Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9), we know that p(a ⊗ b) = psL(a)tL(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For any d ∈ D, we now have Ω(d)(p(a ⊗ b)) = Ω(d)(psL(a)tL(b)) = Ω(d(0))(p)Ψ(d(1))(sL(a)tL(b)) = Ω(d(0))(p)Ψ(d(1))(sL(a))Ψ(d(2))(tL(b)) = Ω(d(0))(p)s′ L(ψ(d(1))(a))t′ L(ψ(d(2))(b)) = Ω(d(0))(p)(ψ(d(1)(a)) ⊗ ψ(d(2)(b))) = Ω(d(0))(p)(ψe(d(1))(a ⊗ b)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='19) (b) Since P and P′ are SAYD modules, it follows from the definition in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9) and the condition in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='13) that ap = ptL(a) a′p′ = p′t′ L(a′) a ∈ A, a′ ∈ A′, p ∈ P, p′ ∈ P′ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='20) where the left hand side of the equalities in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='20) comes from the left A-module action on P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' the left A′-module action on P′) appearing in the structure map ∆P : P −→ U⊳ ⊗A P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' the structure map ∆′ P′ : P′ −→ U′ ⊳ ⊗A′ P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For a ∈ A, u ∈ U and p ∈ P, we now see that d(u ⊗A ap) = Ψ(d(1))(u) ⊗A′ Ω(d(0))(ap) = Ψ(d(1))(u) ⊗A′ Ω(d(0))(ptL(a)) (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='20)) = Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)Ψ(d(1))(tL(a)) = Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)t′ L(ψ(d(1))(a)) = Ψ(d(2))(u) ⊗A′ ψ(d(1))(a)Ω(d(0))(p) (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='20)) = Ψ(d(2))(u) ⊳ ψ(d(1))(a) ⊗A′ Ω(d(0))(p) = t′ L(ψ(d(1))(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p) (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='21) On the other hand, we also have d(u ⊳ a ⊗A p) = d(tL(a)u ⊗A p) (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2)) = Ψ(d(1))(tL(a)u) ⊗A′ Ω(d(0))(p) = Ψ(d(1))(tL(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p) = t′ L(ψ(d(1))(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='22) This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ We are now ready to introduce the notion of a comodule measuring between SAYD modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′ L, t′ L, ∆′ L, ǫ′ L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) be an SAYD-module over U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, a (right) measuring comodule over (C, Ψ, ψ) from P to P′ consists of the following data Ψ : C −→ Vectk(U, U′) ψ : C −→ Vectk(A, A′) Ω : D −→ Vectk(P, P′) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='23) such that (1) (Ψ, ψ) is a measuring of Hopf algebroids from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 14 (2) (Ψ, Ω) is a comodule measuring from the right U-module P to the right U′ module P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (3) For any d ∈ D, the following diagram commutes P ∆P −−−−−−→ U⊳ ⊗A P d:=Ω(d) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� d \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� P′ ∆′ P′ −−−−−−→ U′ ⊳ ⊗A′ P′ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='24) where the right vertical morphism is as defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='18) We will now construct universal measuring comodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By definition, the right comodules over a k-coalgebra C are coalgebras over the comonad ⊗k C : Vectk −→ Vectk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Accordingly, the forgetful functor Comod − C −→ Vectk from the category of right C-comodules has a right adjoint (see, for instance, [7, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4]) that we denote by RC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=', we have natural isomorphisms Vectk(D, V) � Comod − C(D, RC(V)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='25) for any D ∈ Comod − C and V ∈ Vectk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, AL, s′ L, t′ L, ∆′ L, ǫ′ L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) be an SAYD-module over U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let C be a cocommutative coalgebra and (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, there exists a measuring (C, Ψ, ψ)-comodule (QC(P, P′), Θ : QC(P, P′) −→ Vectk(P, P′)) satisfying the following property: given any measuring (C, Ψ, ψ)-comodule (D, Ω : D −→ Vectk(P, P′)) from P to P′, there exists a morphism χ : D −→ QC(P, P′) of right C-comodules such that the following diagram is commutative QC(P, P′) Θ � Vectk(P, P′) D χ �❍❍❍❍❍❍❍❍❍ Ω �t t t t t t t t t t (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We put V := Vectk(P, P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By the adjunction in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='25), there is a canonical morphism ρ(V) : RC(V) −→ V of vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We set QC(P, P′) := � Q, where the sum is taken over all right C-subcomodules over RC(V) such that the restriction ρ(V)|Q : Q −→ V = Vectk(P, P′) is a (C, Ψ, ψ)-comodule measuring from P to P′ in the sense of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It is clear that Θ : ρ(V)|QC(P, P′) : QC(P, P′) −→ V = Vectk(P, P′) is a (C, Ψ, ψ)-measuring comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Additionally, given a measuring (C, Ψ, ψ)-comodule (D, Ω : D −→ Vectk(P, P′)) from P to P′, the adjunction in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='25) gives a morphism χ : D −→ RC(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' But then we notice that ρ(V)|χ(D) : χ(D) −→ V is a measuring (C, Ψ, ψ)-comodule, whence it follows that the image χ(D) ⊆ QC(P, P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The result is now clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U = (U, AL, sL, tL, ∆L, ǫL, S ), U′ = (U′, AL, s′ L, t′ L, ∆′ L, ǫ′ L, S ′) and U′′ = (U′′, A′′ L, s′′ L, t′′ L , ∆′′ L, ǫ′′ L , S ′′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let P, P′ and P′′ be SAYD modules over U, U′ and U′′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Suppose that we have: (1) Ψ : C −→ Vectk(U, U′), ψ : C −→ Vectk(A, A′) and Ω : D −→ Vectk(P, P′) giving the data of a measuring comodule from P to P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (2) Ψ′ : C′ −→ Vectk(U′, U′′), ψ′ : C′ −→ Vectk(A′, A′′) and Ω : D′ −→ Vectk(P′, P′′) giving the data of a measuring comodule from P′ to P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, the following (Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ (Ψ,ψ)⊗(Ψ′,ψ′) −−−−−−−−−−→ V(U, U′) ⊗ V(U′, U′′) −→ V(U, U′′) Ω′ ◦ Ω : D ⊗ D′ Ω⊗Ω′ −−−−→ Vectk(P, P′) ⊗ Vectk(P′, P′′) −→ Vectk(P, P′′) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='27) gives the data of a measuring comodule from P to P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' There is also a canonical morphism of right (C ⊗ C′)-comodules QC(P, P′) ⊗ QC′(P′, P′′) −→ QC⊗C′(P, P′′) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='28) 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We know from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6 that (Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ −→ V(U, U′′) is a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It may also be directly verified that ((Ψ′, ψ′) ◦ (Ψ, ψ), Ω′ ◦ Ω) is a comodule measuring from the right U-module P to the right U′′-module P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' To check the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='24) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6, we observe that for any d ⊗ d′ ∈ D ⊗ D′, u ∈ U and p ∈ P: (d ⊗ d′)(u ⊗A p) = (d ⊗ d′)(1)(u) ⊗A′′ (d ⊗ d′)(0)(p) = d′ (1)(d(1)(u)) ⊗A′′ d′ (0)(d(0)(p)) = d′(d(u ⊗A p))) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='29) Since the measurings (Ψ, ψ, Ω) and (Ψ′, ψ′, Ω′) both satisfy the condition in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='24), it is clear that so does (Ψ′◦Ψ, ψ′ ◦ψ, Ω′ ◦Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Hence, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='27) gives the data of a measuring comodule from P to P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' By definition, QC(P, P′) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' QC′(P′, P′′)) is a measuring comodule from P to P′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' from P′ to P′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='27) it now follows that QC(P, P′) ⊗ QC′(P′, P′′) is a measuring comodule from P to P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The morphism in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='28) is now obtained by the universal property of QC⊗C′(P, P′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ We now consider the “global category of comodules” Comodk whose objects are pairs (C, D), where C is a cocommutative k- coalgebra and D is a right C-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A morphism (f, g) : (C, D) −→ (C′, D′) in Comodk consists of a k-coalgebra morphism f : C −→ C′ and a morphism g : D −→ D′ of C′-comodules, where D is treated as a C′-comodule by corestriction of scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It is clear that putting (C, D) ⊗ (C′, D′) := (C ⊗ C′, D ⊗ D′) makes Comodk into a symmetric monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let S AYDk be the category given by: (a) Objects: pairs (U, P), where U is a Hopf-algebroid and P is an S AYD-module over U (b) Hom-objects: for pairs (U, P), (U′, P′) ∈ S AYDk, we set S AYDk((U, P), (U′, P′)) := (Mc(U, U′), QMc(U,U′)(P, P′)) ∈ Comodk (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='30) Then, S AYDk is enriched over the symmetric monoidal category Comodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For any (U, P) ∈ S AYDk, the scalar multiples of the identity map give a morphism k −→ Mc(U, U) of k-coalgebras, and along with the universal property in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7 give a morphism k −→ QMc(U,U)(P, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now consider (U, P), (U′, P′), (U′′, P′′) ∈ S AYDk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8 with C = Mc(U, U′) and C′ = Mc(U′, U′′), we obtain a morphism QMc(U,U′)(P, P′) ⊗ QMc(U′,U′′)(P, P′) −→ QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) of (Mc(U, U′) ⊗ Mc(U′, U′′))-comodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7, we already have a morphism Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) of k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Combining, we have a morphism S AYDk((U, P), (U′, P′)) ⊗ S AYDk((U′, P′), (U′′, P′′)) −→ (Mc(U, U′′), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='31) in Comodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='31), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) is treated as a Mc(U, U′′)-module via the morphism Mc(U, U′)⊗Mc(U′, U′′) −→ Mc(U, U′′) of k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7, we also know that the morphism Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) arises from the universal property of Mc(U, U′′) applied to the measuring Mc(U, U′) ⊗ Mc(U′, U′′) −→ V(U, U′) ⊗ V(U′, U′′) −→ V(U, U′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Hence, the canonical map QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) −→ Vectk(P, P′′) gives a measuring when treated as a Mc(U, U′′)-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The universal property of QMc(U,U′′)(P, P′′) as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7 now yields a morphism (Mc(U, U′′), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′)) −→ (Mc(U, U′′), QMc(U,U′′)(P, P′′)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='32) in Comodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Composing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='32) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='31), we obtain the required composition of Hom-objects S AYDk((U, P), (U′, P′)) ⊗ S AYDk((U′, P′), (U′′, P′′)) −→ S AYDk((U, P), (U′′, P′′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ 6 Comodule measurings and morphisms on cyclic (co)homology Throughout this section, we fix the following: let U = (U, AL, sL, tL, ∆L, ǫL, S ), and U′ = (U′, A′ L, s′ L, t′ L, ∆′ L, ǫ′ L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let P and P′ be SAYD modules over U and U′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let (Ψ, ψ) : C −→ V(U, U′) be a cocommutative measuring and let Ω : D −→ Vectk(P, P′) be a (C, Ψ, ψ)-measuring comodule from P to P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Since U, U′ are Hopf algebroids, we have recalled in Section 5 that the morphisms β(U) : ◮U ⊗Aop U⊳ −→ U⊳ ⊗A ⊲U and β(U′) : ◮U′ ⊗A′op U′ ⊳ −→ U′ ⊳ ⊗A′ ⊲U′ in the notation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) are bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 16 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For each x ∈ C, the following diagram commutes: U⊳ ⊗A ⊲U β(U)−1 −−−−−−→ ◮U ⊗Aop U⊳ x \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� x \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� U′ ⊳ ⊗A′ ⊲U′ β(U′)−1 −−−−−−→ ◮U′ ⊗A′op U′ ⊳ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) Here, the left vertical map is given by u1 ⊗A u2 �→ x(1)(u1) ⊗A′ x(2)(u2) and the right vertical map by u1 ⊗Aop u2 �→ x(1)(u1) ⊗A′op x(2)(u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It is easy to see that the vertical morphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Further, since β(U) and β(U′) are invertible, it suffices to check that the following diagram commutes ◮U ⊗Aop U⊳ β(U) −−−−−−→ U⊳ ⊗A ⊲U x \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�x ◮U′ ⊗A′op U′ ⊳ β(U′) −−−−−−→ U′ ⊳ ⊗A′ ⊲U′ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2) We now see that for u ⊗Aop v ∈ ◮U ⊗Aop U⊳ and x ∈ C, we have x(β(U)(u ⊗Aop v)) = x(u(1) ⊗A u(2)v) = x(1)(u(1)) ⊗A x(2)(u(2))x(3)(v) = (x(1)(u))(1) ⊗A (x(1)(u))(2)x(2)(v) = β(U′)(x(1)(u) ⊗ x(2)(v)) This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1, it follows in the notation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6) that we have x(1)(u+) ⊗A′op x(2)(u−) = x(u+ ⊗Aop u−) = β(U′)−1(x(u ⊗A 1)) = x(u)+ ⊗A′op x(u)− (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3) for each u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now recall from [14, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1] that the Hochschild homology groups HH•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' the cyclic homology groups HC•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P)) of U with coefficients in the SAYD module P are obtained from the cyclic module C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) := P ⊗Aop (◮U⊳)⊗Aop• with operators as follows (where ¯u := u1 ⊗Aop ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗Aop un, p ∈ P) di(p ⊗Aop ¯u) := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 p ⊗Aop u1 ⊗Aop · · · ⊗Aop un−1tL(ǫL(un)) if i = 0 p ⊗Aop u1 ⊗Aop · · · ⊗Aop un−iun−i+1 ⊗Aop .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' if 1 ≤ i ≤ n − 1 pu1 ⊗Aop u2 ⊗Aop · · · ⊗Aop un if i = n si(p ⊗Aop ¯u) := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 p ⊗Aop u1 ⊗Aop · · · ⊗Aop un ⊗Aop 1 if i = 0 p ⊗Aop · · · ⊗Aop un−i ⊗Aop 1 ⊗Aop un−i+1 ⊗Aop .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' if 1 ≤ i ≤ n − 1 p ⊗Aop 1 ⊗Aop u1 ⊗Aop · · · ⊗Aop un if i = n tn(p ⊗Aop ¯u) := p(0)u1 + ⊗Aop u2 + ⊗Aop · · · ⊗Aop un + ⊗Aop un − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='u1 −p(−1) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4) We now have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For each y ∈ D, the family Ωn(y) : Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) −→ Cn(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) p ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un �→ y(p ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un) = y(0)(p) ⊗ y(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ y(n)(un) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) for n ≥ 0 gives a morphism of cyclic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In particular, we have induced morphisms Ωhoc (y) : HH•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) −→ HH•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) Ωcy (y) : HC•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) −→ HC•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6) on Hochschild and cyclic homologies for each y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From the fact that C is cocommutative and the conditions in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6, it is clear that the morphisms Ωn(y) are well defined, as well as the fact that they commute with the face maps and degeneracies appearing in the cyclic modules C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) and C•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' To verify that the morphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) also commute with the cyclic operators, we note that for p ⊗Aop u1 ⊗Aop ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗Aop un ∈ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) y(tn(p ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un)) = y(p(0)u1 + ⊗ u2 + ⊗ · · · ⊗ un + ⊗ un − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' u1 −p(−1)) = y(0)(p(0))y(1)(u1 +) ⊗ y(2)(u2 +) ⊗ · · · ⊗ y(n)(un +) ⊗ y(n+1)(un −) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='y(2n)(u1 −)y(2n+1)(p(−1)) = y(0)(p(0))y(2)(u1 +) ⊗ y(4)(u2 +) ⊗ · · · ⊗ y(2n)(un +) ⊗ y(2n+1)(un −) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' y(3)(u1 −)y(1)(p(−1)) (since C is cocommutative) = y(0)(p)(0)y(1)(u1 +) ⊗ y(3)(u2 +) ⊗ · · · ⊗ y(2n−1)(un +) ⊗ y(2n)(un −) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' y(2)(u1 −)y(0)(p)(−1) (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='24)) = y(0)(p)(0)y(1)(u1)+ ⊗ y(2)(u2)+ ⊗ · · · ⊗ y(n)(un)+ ⊗ y(n)(un)− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' y(1)(u1)−y(0)(p)(−1) (using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3)) This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ We now come to cyclic cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For this, we recall that from [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1] that the Hochschild cohomology groups HH•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' the cyclic cohomology groups HC•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P)) of U with coefficients in the SAYD module P are obtained from the cocyclic module C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) := (⊲U⊳)⊗A• ⊗A P with operators as follows (where ¯u := u1 ⊗A ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗A un, p ∈ P) δi(¯u ⊗A p) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 1 ⊗A u1 ⊗A · · · ⊗A un ⊗A p if i = 0 u1 ⊗A · · · ⊗A ∆L(ui) ⊗A · · · ⊗A un ⊗A p if 1 ≤ i ≤ n u1 ⊗A · · · ⊗A un ⊗A p(−1) ⊗A p(0) if i = n + 1 δi(p) = � 1 ⊗A p if j = 0 p(−1) ⊗A p(0) if j = 1 σi(¯u ⊗A p) = u1 ⊗A · · · ⊗A ǫL(ui+1) ⊗A · · · ⊗A un ⊗A p 0 ≤ i ≤ n − 1 τn(¯u ⊗A p) = u1 −(1)u2 ⊗A · · · ⊗A u1 −(n−1)un ⊗A u1 −(n)p(−1) ⊗A p(0)u1 + (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7) We now have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For each y ∈ D, the family Ω n(y) : Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) −→ Cn(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un ⊗ p �→ y(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un ⊗ p) = y(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ y(n)(un) ⊗ y(0)(p) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8) for n ≥ 0 gives a morphism of cocyclic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' In particular, we have induced morphisms Ω hoc(y) : HH•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) −→ HH•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) Ω cy(y) : HC•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) −→ HC•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P′) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='9) on Hochschild and cyclic cohomologies for each y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' It is clear that the morphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For y ∈ D and i = n + 1 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7), we note that y(δn+1(u1 ⊗ · · · ⊗ un ⊗ p)) = y(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' y(n)(un) ⊗ y(n+1)(p(−1)) ⊗ y(0)(p(0)) = y(2)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' y(n+1)(un) ⊗ y(1)(p(−1)) ⊗ y(0)(p(0)) (since C is cocommutative) = y(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' y(n)(un) ⊗ (y(0)(p))(−1) ⊗ y(0)(p)(0) (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='24)) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='10) Similarly, we may verify that the morphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='8) commute with the face and degeneracy maps appearing in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' To show that they also commute with the cyclic operators appearing in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='7), we note that for u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un ⊗ p ∈ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) and y ∈ D, we have y(τn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' ⊗ un ⊗ p)) = y(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(1)u2 ⊗A · · · ⊗A u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(n−1)un ⊗A u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(n)p(−1) ⊗A p(0)u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='= y(1)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(1))y(2)(u2) ⊗A · · · ⊗A y(2n−3)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(n−1))y(2n−2)(un) ⊗A y(2n−1)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(n))y(2n)(p(−1)) ⊗A y(0)(p(0)u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='= y(1)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(1))y(n+1)(u2) ⊗A · · · ⊗A y(n−1)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(n−1))y(2n−1)(un) ⊗A y(n)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−(n))y(2n)(p(−1)) ⊗A y(0)(p(0)u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='= y(1)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−)(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−)(n−1)y(n)(un) ⊗A y(1)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−)(n)y(n+1)(p(−1)) ⊗A y(0)(p(0)u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='= y(2)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−)(1)y(3)(u2) ⊗A · · · ⊗A y(2)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−)(n−1)y(n+1)(un) ⊗A y(2)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='−)(n)y(n+2)(p(−1)) ⊗A y(0)(p(0))y(1)(u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='= y(1)(u1)−(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1)−(n−1)y(n)(un) ⊗A y(1)(u1)−(n)y(n+1)(p(−1)) ⊗A y(0)(p(0))y(1)(u1)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='(using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3)) = y(1)(u1)−(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1)−(n−1)y(n)(un) ⊗A y(1)(u1)−(n)y(0)(p)(−1) ⊗A y(0)(p)(0)y(1)(u1)+ (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='24)) This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ 18 Finally, we recall from [14, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3] that there are Hopf-Galois isomorphisms relating the modules C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) and C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) : Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) � −→ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) p ⊗ u1 ⊗ · · · ⊗ un �→ u1 (1) ⊗ u1 (2)u2 (1) ⊗ · · · ⊗ u1 (n)u2 (n−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='un−1 (2) un ⊗ p (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11) We will conclude this section by showing that the morphisms induced by comodule measurings of SAYD modules are compat- ible with the Hopf-Galois isomorphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let U = (U, AL, sL, tL, ∆L, ǫL, S ), and U′ = (U′, A′ L, s′ L, t′ L, ∆′ L, ǫ′ L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let P and P′ be SAYD modules over U and U′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Let (Ψ, ψ) : C −→ V(U, U′) be a cocommutative measuring and let Ω : D −→ Vectk(P, P′) be a (C, Ψ, ψ)-measuring comodule from P to P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Then, for each y ∈ D, the following diagram commutes Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='P) −−−−−−→ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) Ωn(y) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6� \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�Ω n(y) Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='P) −−−−−−→ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We set N := n(n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' For y ∈ D and p ⊗ u1 ⊗ · · · ⊗ un ∈ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P), we see that Ω n(y)(ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P)(p ⊗ u1 ⊗ · · · ⊗ un)) = Ω n(y)(u1 (1) ⊗ u1 (2)u2 (1) ⊗ · · · ⊗ u1 (n)u2 (n−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' un−1 (2) un ⊗ p) = y(1)(u1 (1)) ⊗ y(2)(u1 (2))y(3)(u2 (1)) ⊗ · · · ⊗ y(N+1)(u1 (n))y(N+2)(u2 (n−1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' y(N+n−1)(un−1 (2) )y(N+n)(un) ⊗ y(0)(p) = y(1)(u1 (1)) ⊗ y(2)(u1 (2))y(n+1)(u2 (1)) ⊗ · · · ⊗ y(n)(u1 (n))y(2n−1)(u2 (n−1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='y(N+n−1)(un−1 (2) )y(N+n)(un) ⊗ y(0)(p) = y(1)(u1)(1) ⊗ y(1)(u1)(2)y(2)(u2)(1) ⊗ · · · ⊗ y(1)(u1)(n)y(2)(u2)(n−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='y(n−1)(un−1)(2)y(n)(un) ⊗ y(0)(p) = ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' P)(Ωn(y)(p ⊗ u1 ⊗ · · · ⊗ un)) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='13) □ 7 Operads with multiplication, comp modules and morphisms on cyclic homology We start the final section by recalling from Kowalzig [16] the following two notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (see [16, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2]) A non-Σ operad O over k consists of the following: (a) A collection of vector spaces O = {O(n)}n≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (b) A family of k-linear operations ◦i : O(p)⊗O(q) −→ O(p+q−1) and an identity 1 ∈ O(1) satisfying the following conditions (for φ ∈ O(p), ψ ∈ O(q), χ ∈ O(r)) φ ◦i ψ = 0 if p < i or p = 0 (φ ◦i ψ) ◦ j χ = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 (φ ◦ j χ) ◦i+r−1 ψ if j < i φ ◦i (ψ ◦ j−i+1 χ) if i ≤ j < q + i (φ ◦ j−q+1 χ) ◦i ψ if j ≥ q + i φ ◦i 1 = 1 ◦1 φ = φ for i ≤ p (c) An operad multiplication µ ∈ O(2) and a unit e ∈ O(0) such that µ ◦1 µ = µ ◦2 µ µ ◦1 e = µ ◦2 e = 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1) Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (see [16, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='1]) A cyclic unital comp module M over an operad O with multiplication consists of the following data: (a) A collection of vector spaces M = {M(n)}n≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' 19 (b) A family of k-bilinear operations •i : O(p) ⊗ M(n) −→ M(n − p + 1), 0 ≤ i ≤ n + 1 − p satisfying the following conditions for φ ∈ O(p), ψ ∈ O(q), x ∈ M(n) φ •i (ψ • j x) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 ψ • j (φ •i+q−1 x) j < i (φ • j−i+1 ψ) •i x if j − p < i ≤ j ψ • j−p+1 (φ •i x) if 1 ≤ i ≤ j − p as well as 1 •i x = x for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (c) A cyclic operator t : M(n) −→ M(n) for n ≥ 1 satisfying t(φ •i x) = φ •i t(x) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='2) for φ ∈ O(p), x ∈ M(n) and 0 ≤ i ≤ n − p as well as tn+1 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We take pairs (O, M) consisting of a non-linear Σ operad O and a cyclic unital comp module M over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now consider comodule measurings between such pairs Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' A comodule measuring from (O, M) to (O′, M′) consists of the following: (a) A cocommutative coalgebra C and a family of morphisms {Φn : C −→ Vectk(O(n), O′(n))}n≥0 satisfying Φp+q−1(x)(φ ◦i ψ) = Φp(x(1))(φ) ◦′ i Φq(x(2))(φ) Φ2(x)(µ) = ǫ(x)µ′ Φ0(x)(e) = ǫ(x)e′ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3) for φ ∈ O(p), ψ ∈ O(q) and any x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' (b) A comodule D over C and a family of morphisms {Ψn : D −→ Vectk(M(n), M′(n))}n≥0 satisfying Ψn−p+1(φ •i x) = Ψp(y(0))(φ) •i Ψn(y(1))(x) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4) for y ∈ D, φ ∈ O(p), x ∈ M(n), 0 ≤ i ≤ n + 1 − p and also Ψn(y)(t(x)) = t′(Ψn(y)(x)) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) for y ∈ D, x ∈ M(n), where t and t′ are respectively the cyclic operators on M and M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We now recall from [16, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5] that the cyclic homology of (O, M) is obtained from the cyclic module C•(O, M) := M(•) whose cyclic operators are t : M(n) −→ M(n) and whose face maps and degeneracies are given as follows: di(x) := µ •i x, (0 ≤ i < n) dn(x) := µ •0 t(x) sj(x) := e • j+1 x, 0 ≤ j ≤ n (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6) The cyclic homologies of this cyclic module will be denoted by HC•(O, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We conclude with the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' If D is a C-measuring comodule from (O, M) to (O′, M′), then each y ∈ D induces a morphism Ψcy (y) : HC•(O, M) −→ HC•(O′, M′) on Hochschild homologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' We know from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='5) that the action of any y ∈ D commutes with the cyclic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' From the definitions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='6) and the conditions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content='3), it is clear that the action also commutes with the degeneracies and face maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' The result is now clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'} +page_content=' □ References [1] M.' metadata={'source': 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