diff --git "a/DdE1T4oBgHgl3EQfWQSA/content/tmp_files/load_file.txt" "b/DdE1T4oBgHgl3EQfWQSA/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/DdE1T4oBgHgl3EQfWQSA/content/tmp_files/load_file.txt" @@ -0,0 +1,739 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf,len=738 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='03112v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='AT] 8 Jan 2023 PERIODIC CYCLIC HOMOLOGY OVER Q KONRAD BALS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let X be a derived scheme over an animated commutative ring of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We give a complete description of the periodic cyclic homology of X in terms of the Hodge completed derived de Rham complex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular this extends earlier computations of Loday-Quillen to non-smooth algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, we get an explicit condition on the Hodge completed derived de Rham complex, that makes the HKR-filtration on periodic cyclic homology constructed by Antieau and Bhatt-Lurie exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Introduction For a commutative ring k and a k-algebra R, the Hochschild homology HH(R/k) gives an element in the derived category D(k) of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It has proven itself to be an interesting invariant, appearing for example in trace methods computing algebraic K-theory or in Connes’ non-commutative geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It was also Connes in [Con85] who constructed the cyclic structure on Hochschild homology to define negative cyclic homology HC−(R/k) := HH(R/k)hS1 and later periodic cyclic homology HP(R/k) := HH(R/k)tS1 and proving a relation between HC− of smooth functions on a manifold and de Rham cohomology of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Transferring Connes’ geometric interpretation into algebraic observations in [LQ84] Loday and Quillen compute the homotopy groups HC− ∗ (R/k) in terms of algebraic de Rham cohomology in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For the purpose of this paper passing here to the Tate-construction, they prove: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1 ([LQ84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Assume Q ⊂ k commutative and R a smooth commutative k-algebra, then HP∗(R/k) ∼= � n∈Z H∗−2n dR (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' k) In this paper we give a generalization of this computation to the non-smooth and non-affine situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the classical observation that HH(R[S−1]/k) ≃ HH(R/k) ⊗R R[S−1] for every affine open Spec(R[S−1]) ⊂ SpecR Hochschild homology extends to a sheaf HHk in the Zariski1 topology on schemes over k (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [WG91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact, similarly we get a sheaf HPk extending periodic cyclic homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We recall the details in Appendix A and write HH(X/k) := Γ(X, HHk) and HP(X/k) := Γ(X, HPk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, Hochschild homology as a functor CAlg♥ k → D(k) from discrete k-algebras to D(k) is left Kan extended from discrete polynomial algebras2 and, thus, further extends to a sifted colimit preserving functor from the category of animated commutative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' simplicial commutative) k- algebras CAlgan k/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' So putting both generalizations together and writing LΩ∗ X/k for the derived de Rham complex of a derived scheme X over an animated Q-algebra k, we can state our main theorem in a great generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, if k is discrete and X = Spec(R) for a discrete k-algebra R, this gives new results on the periodic cyclic homology of ordinary algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given an animated commutative ring k with Q ⊂ π0(k) and X a derived k-scheme, we have HP(X/k) ≃ � n∈Z � LΩ∗ X/k[−2n] 1In fact by [BMS19] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' even in the fpqc topology via a different argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 2If P• is a simplicial resolution of the k-algebra R, it suffices to check that | HH(P•/k)| ≃ HH(R/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1 2 KONRAD BALS where � LΩ∗ X/k is the completion of LΩ∗ X/k with respect to the Hodge filtration LΩ≥• −/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The key ingredient in the proof is to understand how the Tate-construction behaves under the passage from smooth algebras to general or even animated algebras and it is this behavior that lets the product appear on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In [Ant19] Antieau constructs the HKR-filtration on HP(X/k) with n-th associated graded � LΩ∗ X/k[2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If k is an (animated) Q-algebra we can give a complete identification of this HKR- filtration in terms of the equivalence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 and we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 the HKR-filtration on HP(X/k) corresponds to the ascending partial product filtration on � n∈Z � LΩ∗ X/k[−2n], that is Fili HKR HP(X/k) ≃ � n≤−i � LΩ∗ X/k[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, the HKR-filtration is exhaustive, if and only if � LΩ∗ X/k is (homologically) bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This criterion will give us a large class of examples with exhaustive HKR-filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If k is a discrete Noetherian commutative Q-algebra and X an ordinary scheme of finite type over Spec k, then Bhatt gives in [Bha12] a concrete way to compute � LΩ∗ X/k, which in particular lives in non- positive degrees (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Passing to filtered colimits we get Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If k is a discrete Q-algebra and X an ordinary qcqs scheme over k, then the HKR- filtration on HP(X/k) is exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Furthermore, the analysis of the Tate-filtration in characteristic 0, which is reviewed in the Appendix B, also gives a description of the multiplicativity of the equivalence in the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In general for an algebra A ∈ CAlgk there is just no algebra structure on � n∈Z A[−2n], however, for a (animated) commutative k-algebra R, the object HP(R/k) carries a natural structure of a commutative algebra in Modk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In section 4 we construct the corresponding multiplicative structure on � n∈Z � LΩ∗ X/k[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' On homotopy groups the induced graded ring structure comes from LΩ≤n X/k((t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Note that there is the terminal topology on π∗ � LΩ∗ X/k making the maps π∗ � LΩ∗ X/k → π∗LΩ≤n X/k continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It is not Hausdorff because not every element is detected in some π∗LΩ≤n R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' With this we can almost completely describe the graded ring π∗ HP(X/k) in terms of π∗ � LΩ∗ X/k: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' we can describe the homotopy groups HP∗(X/k) algebraically as HP∗(X/k) ∼= �� n∈Z antn : an ∈ π∗+2n � LΩ∗ X/k � with addition and multiplication given as �� n∈Z antn � + �� n∈Z bntn � = � n∈Z (an + bn)tn �� n∈Z antn � �� n∈Z bntn � = � n∈Z cntn where cn is a limit of the finite partial sums of � i+j=n ai · bj in the topology on π∗ � LΩ∗ R/k 3 3This is sometimes called a net and explicitly means for every open U ∋ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' there is a finite subset I0 ⊂ {i+j = n},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' such that for all finite subset J ⊂ {i + j = n} containing I0 we have cn − � (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='j)∈J ai · bj ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' PERIODIC CYCLIC HOMOLOGY OVER Q 3 However, we want to immediately issue the warning that because the topology on π∗ � LΩ∗ R/k is not Hausdorff, the element cn ∈ π∗ � LΩ∗ R/k is not uniquely determined as a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' To fully understand the homotopy groups HP∗(X/k) algebraically, one, furthermore, has to analyze the lim1-terms contributing to π∗ � LΩ∗ R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We begin in section 2 with a formality statement for S1-actions in the derived cat- egory over rational algebras (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3) in order to recall a coherent version of the HKR-theorem in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This allows us to coherently compute HP for smooth algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In section 3 we will use the language of filtrations in order to generalize the computations for smooth algebras to arbitrary derived schemes and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particu- lar we will use the multiplicativity of the Tate-filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The Tate-filtration itself and its multiplic- ative structure in the rational setting will be reviewed in the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Furthermore in section 3 we will exploit the consequences for the HKR-filtration and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, the last section (4) is completely devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Throughout this note we are freely using the ∞-categorical language as developed in [Lur09] and [Lur16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, for a commutative ring k we identify the derived category D(k) with the category Modk := ModHkSp of Hk-module spectra and thus view it as a stably symmetric monoidal ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It comes with a canonical lax symmetric monoidal functor ι: Ch∗(k) → D(k) from the 1-category of chain complexes and we will constantly abuse notation by identifying C∗ with ιC∗ for C∗ ∈ Ch∗(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, we will use the 1-category CDGAk of commutative differential graded algebras over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' An object (C∗, d) ∈ CDGAk consists of a commutative graded k-algebra � i∈Z Ci of discrete R-modules with differentials d: Ci−1 → Ci for all i > 0 satisfying the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' There will be two orthogonal ways to view a CDGAk as an object in CAlgk, either with 0 differential or with differential d and we already warn the reader to not confuse those functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, for a commutative ring k and a commutative k-algebra R, we will generally view the de Rham complex Ω∗ R/k as an object in CAlgk := CAlg(Modk), and if we want to view it as a CDGA over k we write ΩH R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Later in the paper, we need to talk about filtrations in a stable category C, by which we always mean decreasingly indexed, Z-graded filtrations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' functors from Zop ≤ into C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a symmetric monoidal category C we equip the category Fil(C) := Fun(Zop ≤ , C) with the Day convolution tensor product ⊗Day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The n-th associated graded grnF of F is given by the cofibre of the map F n+1 → F n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' A splitting of a filtration F • ∈ Fil(C) consists of a collection (An)n∈Z together with an map of filtrations � n≥• An → F • inducing an equivalence on associated graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, a splitting (An) of F canonical gives an identification grnF ≃ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, to fix vocabulary, a filtration F • ∈ Fil(C) on F ∈ C is complete if lim F • ≃ 0 and is exhaustive if colim F • ≃ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We write Fil∧(C) ⊂ Fil(C) for the full subcategory on complete filtrations and denote by (−)∧ its left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' I would like to thank Achim Krause, Jonas McCandless and Thomas Nikolaus for helpful discussions on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, again I want to thank Thomas Nikolaus for bringing this project up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure and the CRC 1442 Geometry: Deformations and Rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Formality over Q The explicit computations heavily rely on strong formality properties that hold if working over Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In this section we will prove a strong version of the HKR-theorem for Hochschild homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This enables us to establish a coherent versions of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1 copied from [LQ84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 4 KONRAD BALS Throughout the first section, let k be a discrete commutative Q-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The key ingredient is a formality statement of C∗(S1, k), due to [TV11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The multiplication S1 × S1 → S1 and the diagonal S1 → S1 × S1 exhibit S1 as an associative bialgebra in spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because the symmetric monoidal structure on S is cartesian, by the dual of [Lur15][Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8] the coalgebra structure given by the diagonal refines to a cocommutative coalgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now the functor C∗(−, k): S → D(k) from spaces to the derived category of k taking singular chains with coefficients in k refines via the Eilenberg- Zilber maps to a symmetric monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Therefore C∗(S1, k) acquires the structure of a cocommutative bialgebra in D(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, the functor ι: Ch∗(k) → D(k) from the 1-category of chain complexes to the ∞- category D(k) is lax symmetric monoidal and precisely restricts to a symmetric monoidal functor on the full 1-subcategory ChK−flat ∗ (k) of K-flat chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus the chain complex for ǫ in degree 1 Λk(ǫ) := (k · ǫ 0−→ k · 1) with multiplication ǫ2 = 0 and primitive comultiplication ∆ǫ = (ǫ⊗1+1⊗ǫ) gives a cocommutative bialgebra object in D(k) under the identification of Λk(ǫ) as an element in D(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 ([TV11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In this setting where k is a discrete Q-algebra, there is a natural equi- valence C∗(S1, k) ≃ Λk(ǫ) as cocommutative bialgebras in D(k) for ǫ primitive in degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For completeness reasons we would like to include a proof here: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Both objects C∗(S1, k) and Λk(ǫ) have canonical augmentations coming from S1 → ∗ in S and ǫ �→ 0 in Ch∗(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will in fact show, that they even agree as augmented cocommutative algebras in D(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Using the adjunction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [Lur16] Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='174) bar: Algaug(coCAlgD(k)) coCAlgaug(D(k)) :cobar it satisfies to construct a map of (co-)augmented cocommutative coalgebras under the bar-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact the computation in [Ada56] show that for C∗(S1, k) the unit of the adjunction C∗(S1, k) → cobar(barC∗(S1, k)) is an equivalence, so that an identification of barC∗(S1, k) ≃ C∗(BS1, k) trans- lates to an identification of C∗(S1, k) under cobar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Therefore, we want to understand the cocommut- ative coalgebra structure of barC∗(BS1, k) or equivalently the dual commutative algebra structure on C∗(BS1, k), as both objects are of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' A choice of a generator in H2(BS1, k) gives a map k[x] := Free(k[−2]) → C∗(BS1, k) from the free commutative k-algebra on a generator x in degree −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because Q ⊂ k, on homotopy groups both sides are free on a generator in degree −2 and we have C∗(BS1, k) ≃ k[x] is free as a commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, translating back to cocommutative bialgebras, we can compute cobar(k[x])∨ ≃ (bark[x])∨ ≃ � k ⊗k[x] k �∨ by resolving k with the DGA (Λk[x](ǫ∨), dǫ∨ = x) for a primitive element ǫ∨ in degree −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus � k ⊗k[x] k �∨ ≃ Λk(ǫ∨)∨ ≃ Λk(ǫ) for ǫ a dual basis to ǫ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ From now on, to shorten notation we set A := Λk(ǫ) for |ǫ| = 1 primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a rational discrete algebra k the categories Fun(BS1, D(k)) and ModAD(k) are equivalent as symmetric monoidal categories, where the symmetric monoidal structure on the latter comes from the coalgebra structure on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 4There is a gap in the proof of the cited reference as pointed out by [DH22], which could be fixed in the latest version (v4) of [BCN21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' PERIODIC CYCLIC HOMOLOGY OVER Q 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' There is a symmetric monoidal equivalence Fun(BS1, D(k)) ≃ ModC∗(S1,k)D(k) as sym- metric monoidal categories, where the symmetric monoidal structure on the right hand side comes from the cocommutative bialgebra structure on C∗(S1, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus the equivalence C∗(S1, k) ≃ A as cocommutative bialgebras gives a symmetric monoidal equivalence of their module categories (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' in [Rak20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The above equivalence induces the identity on underlying objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, given a complex X ∈ D(k) equipping X with an action of S1 is equivalent to providing a module structure over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Informally, this amounts to a map d: k · ǫ[1] ⊗ X ≃ X[1] → X and coherent homotopies witnessing d2 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let CDGAk denote the 1-category of commutative differential graded algebras over k as introduced in the Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Forgetting the differential, there is a functor CDGAk → CAlgCh∗(k) sending (C∗, d) ∈ CDGA to � i∈Z Ci[i] ∈ CAlgCh∗(k) with 0 differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In 1- categories now an action of A precisely corresponds to an ascending differential, such that this functor refines through CAlgModACh∗(k) and postcomposing with ι we get a map CDGAk → CAlgModACh∗(k) → CAlgModAD(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' To avoid confusion we will write U : CDGAk → CAlgBS1 k for this functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a k-algebra R the de Rham complex ΩH R/k by definition lives in CDGAk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now via the previous construction the underlying chain complex UΩH R/k ≃ � n∈N Ωn R/k[n] ≃ � · · 0−→ Ω2 R/k 0−→ Ω1 R/k 0−→ Ω0 R/k � (1) gives an object in CAlgBS1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This simplifies the analysis of Hochschild homology in the rational setting and we can phrase a strong version of the HKR-theorem, which has been well known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [Qui70]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, we would like to emphasize on all the structure the following result captures and give a different proof as in the cited source: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If Q ⊂ k, then for every smooth discrete k-algebra R, there are natural equival- ences HH(R/k) ∼ −→ UΩH R/k ≃ � n∈N Ωn R/k[n] of commutative algebras in D(k) with S1-action, where the S1-action on the right hand side is given by the de Rham differential (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the category CAlgBS1 k Hochschild homology enjoys a universal property: For every com- mutative k-algebra S with S1-action any non-equivariant map R → S extends uniquely up to contractible choice over R → HH(R/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus we get the dashed S1-equivariant algebra map R ≃ Ω0 R/k � n∈N Ωn R/k[n] HH(R/k) The original computation in the HKR-theorem [HKR62] gives an equivalence ΩH R/k ∼ −→ HH∗(R/k) of differentially graded algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Postcomposing with the map above on homotopy groups, we get a map ΩH R/k → HH∗(R/k) → ΩH R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, ΩH R/k has a universal property among commutative differentially graded algebras, as the initial CDGA with a map from R into its zeroth part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because on the zeroth part the composition above is given by the identity R → R, the same is true for the entire map, forcing HH∗(R) → ΩH R/k to be an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ 6 KONRAD BALS Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This strong version of the HKR-theorem can be understood as a rigidification of the Hochschild homology functor from polynomial k-algebras Polyk: It gives a functorial factorization CDGAk Polyk CAlgModAD(k), U ΩH −/k HH(−/k) through the functor ΩH −/k : Polyk → CDGAk of 1-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We can now get a very good understanding of the Tate-construction for such formal objects: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For (C∗, d) ∈ CDGAk we write |C∗| for the chain algebra (C−∗, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This gives a functor |−|: CDGAk → CAlgCh∗(k) of 1-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' More generally, given a graded object C∗ with differential d we want to write |C∗| to stress that we view it as a chain complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By definition we have |ΩH R/k| ≃ Ω∗ R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' With this notation set, we can make the classical computations of periodic cyclic homology in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This is also done for example in the lectures [KN18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For (C∗, d) ∈ CDGAk there is a natural map |C∗| → (UC∗)tS1 in CAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because of the lax monoidal natural transformation (−)hS1 → (−)tS1, it suffices to estab- lish a natural map |C∗| → ChS1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Under the symmetric monoidal equivalence Fun(BS1, D(k)) ≃ ModAD(k) the functor (−)hS1 corresponds to mapA(k, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' A choice of projective resolution P∗ of k as an A-coalgebra reduces us to give a functorial map |C∗| → mapA(P∗, UC∗) where the right hand side is the 1-categorical mapping chain complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now put P∗ = (A⟨t∨⟩, dP ) as the free divided power algebra on a primitive generator t∨ in degree 2 with dP t∨ = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, computing the mapping chain complex gives an equivalence mapA(P∗, UC∗) ∼= (UC∗�t�, td) for |t| = −2 a dual generator to t∨ and we can explicitly describe a multiplicative chain map |Ci| → (UC∗�t�, td) given by Ci ≃ Ci · ti → UC∗�t� on chain groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The computation in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11 actually completely describes UChS1 ∗ and under the equivalence UCtS1 ∗ ≃ UChS1 ∗ ⊗khS1 ktS1 we already get a full identification UCtS1 ∗ ≃ (UC∗((t)), td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For C∗ = ΩH R/k for R smooth over a rational algebra k we thus could have a full understanding of HP(R/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, we will not directly use this, but rather proof a general statement with more structure, that generalizes to non-smooth and animated algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Main Theorem Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given C∗ ∈ CDGAk, we denote by Fil• H|C∗| the filtration Filn H|C∗| := |τ≥nC∗| where τ≥nC∗ is the part of grading greater or equal n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Unraveling, Fil• H|C∗| precisely gives the stupid or brutal filtration on the chain complex |C∗| ∈ Ch∗(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, for X ∈ Fun(BS1, Sp) let Fil• T XtS1 be the Tate filtration on XtS1, see Appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It is a complete commutatively multiplicative and exhaustive filtration with associated graded grnFilT XtS1 ≃ X[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The Tate-filtration also restricts to a complete (and exhaustive) filtration on Fil0 T XtS1 ≃ XhS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' PERIODIC CYCLIC HOMOLOGY OVER Q 7 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For C∗ ∈ CDGAk the map |C∗| → UCtS1 ∗ refines and extends to an equivalence � Fil• H|C∗| ⊗Day Fil• T ktS1�∧ → Fil• T UCtS1 ∗ of commutatively multiplicative filtered objects in D(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the concrete description of ChS1 ∗ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11, we can identify the Tate-filtration with the t-adic filtration on (C∗�t�, td) via Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='12 and the map from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11 refines to a map of commutatively multiplicative filtrations Fil• H|C∗| → Fil• T UCtS1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because the target is a module over the commutative algebra Fil• T ktS1, we get the map Fil• H|C∗| ⊗Day Fil• T ktS1 → Fil• T UCtS1 ∗ (2) and because the target is complete, it even factors over the completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' To show that we get an equivalence of complete filtrations, it is enough to check on associated graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let us introduce a formal character t in degree −2 to visually relate Tate filtrations and t-adic filtrations and write grnFil• T ktS1 ≃ k[−2n] =: k · tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Then on the nth associated graded the map (2) is given by � i+j=n Ci[−i] ⊗ k · tj ≃ � i+j=n Ci[i] · ti ⊗ k · tj → UC∗ · tn and thus an equivalence by construction, as UC∗ ≃ � Ci[i] as a complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ We can finally return to our situation of interest and immediately get a description of HP(R/k) in more general situations: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If k is an animated ring with rational homotopy groups and R in (CAlgan)k/, then there is an equivalence of commutatively multiplicative complete filtrations � Fil• HLΩ∗ R/k ⊗Day Fil• T ktS1�∧ → Fil• T HP(R/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' (3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' First assume that k is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We want to show that both sides commute with sifted colimits as functors to Fil∧(D(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For Fil• HLΩ∗ R/k after completion this is by definition and because the Day convolution tensor product commutes with all colimits it follows for the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' As functors to complete filtrations we can also check this on associated graded for the right hand side: And also here any shifts of HH(R/k) commute with sifted colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We thus can reduce to the case that k is an ordinary Q-algebra and R smooth over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Then the equivalence immediately follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 by putting C∗ = ΩH R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the general case of an animated morphism k → R between animated Q-algebras we can give the exact same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Choose a simplicial resolution kn → Rn of polynomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Again by definition Fil• HLΩ∗ R/k ≃ colim Fil• HLΩ∗ Rn/kn and thus the left hand side is determined by its value on polynomial rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' On the right hand side we check again, that on associated graded we get an equivalence HH(R/k) ≃ HH(R/Q) ⊗HH(k/Q) k ≃ colim HH(Rn/Q) ⊗HH(kn/Q) kn where the first equivalence comes from the base-change formula for Hochschild homology (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [AMN18] proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4) and the second from the facts that HH(−/Q) commutes with colimits in CAlgQ and that the colimit is sifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus also in the general case, the statement reduces to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Finally, in order to compute the periodic cyclic homology in our case, we only have to understand the left hand filtration in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' There are basically two obstacles, that we have to take care of: Completion does not behave well with Day convolution and does not behave well with underlying objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 8 KONRAD BALS Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let k be an animated ring with Q ⊂ π0k and X a derived scheme over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Then there is a natural equivalence of underlying objects in Modk HP(X/k) ≃ � n∈Z � LΩ∗ X/k[−2n] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because both sides are sheaves in the Zariski topology on X we are reduced to the case X = SpecR for R ∈ CAlgan k/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the above Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 there is a natural equivalence of filtrations � Fil• HLΩ∗ R/k ⊗Day Fil• T ktS1�∧ → Fil• T HP(R/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because the Tate-filtration is exhaustive on HP(R/k) it suffices to compute the underlying object of the left filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now the filtration FilT ktS1 carries a canonical splitting, because the connecting homomorphism in Filn+1 T ktS1 Filn T ktS1 grnFil• T ktS1 khS1[−2(n + 1)] khS1[−2n] k[−2n] is forced to vanish for degree reasons, in fact Map(k[−2], khS1[−2n − 3]) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Therefore, we have a map of filtrations � n≥• k[−2n] → Fil• T ktS1, inducing an equivalence on associated graded, and, thus, as the left hand side is complete, it even is an equivalence of filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We claim now, that this splitting induces an equivalence � n∈Z (Fil•−n H LΩ∗ R/k[−2n])∧ ≃ (FilHLΩ∗ R/k ⊗Day FilT ktS1)∧ Indeed, the canonical map � n∈Z(Fil•−n H LΩ∗ R/k[−2n]) → � n∈Z(Fil•−n H LΩ∗ R/k[−2n])∧ exhibits the right hand side as the completion: It is evidently complete and the map on the m-th associated graded � n∈Z LΩm−n R/k [−2n] → � i∈Z LΩm−n R/k [−2n] is an equivalence, because LΩm−n R/k is always bounded below and 0 for n > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally we want to compute the underlying object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' the colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Consider the canonical colimit-limit-interchange can map sitting in the cofibre sequence colim �� n∈Z Fil•−n H � LΩ∗ R/k[−2n] � can −−→ � n∈Z � LΩ∗ R/k[−2n] → colim �� n∈Z LΩ≤•−n−1 R/k [−2n] � But because LΩ≤•−n−1 R/k is bounded below for all n, and 0 for n ≥ •, the right most product is actually degreewise finite and, thus, vanishes in the colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now putting everything together gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ We want to use the result to investigate the exhaustiveness of the HKR-filtration constructed in [Ant19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It arises from the left Kan extension of the Beilinson Whitehead tower of the Tate filtration on HP(−/k) from smooth algebras to bicomplete filtrations as the underlying outer filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For more details c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' or [BL22] section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4, the HKR-filtration on HP(R/k) can be identified with the filtration by partial products of � n∈Z � LΩ∗ Rn/kn[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Precisely, Fili HKR HP(R/k) ≃ � n≤−i � LΩ∗ Rn/kn[−2n] PERIODIC CYCLIC HOMOLOGY OVER Q 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By definition of the HKR-filtration we only have to construct equivalences in the case R over k a smooth algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' But now the Tate-filtration on HP(R/k) induces a shifted Hodge filtration on the factor Ω∗ R/k[2n] with Film T (Ω∗ R/k[2n]) ≃ (Filn+m H Ω∗ R/k)[2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because Filn+m H Ω∗ R/k ∈ D(k)≤−n−m we have Film T (Ω∗ R/k[2n]) ∈ D(k)≤n−m Moreover, we can similarly compute grmFil• T (Ω∗ R/k[2n]) ≃ Ωn+m R/k [−n − m + 2n] ∈ D(k)≥n−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact these two conditions precisely show that Fil• T (Ω∗ R/k[2n]) is concentrated in degree n with respect to the Beilinson t-structure on Fil(D(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' From our complete description of Fil• T HP(R/k) in terms of Ω∗ R/k · [2n] we get Fil• T (Ω∗ R/k[2n]) ≃ πnFil• T HP(R/k) ≃ grnFil• HKR HP(R/k) where the last equivalence comes form the definition of the HKR-filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, HP(R/k) decomposes into the product of the associated gradeds of the HKR-filtration, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of the theorem the HKR-filtration from [Ant19] is exhaustive if and only if � LΩ∗ X/k is bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We can phrase the exhaustiveness as the condition that the natural map colim i � n≥i � LΩ∗ X/k[2n] → � n∈Z � LΩ∗ X/k[2n] ≃ � n∈Z � LΩ∗ X/k[−2n] is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This is precisely the case when � LΩ∗ X/k[2n] eventually leaves any fixed degree for n → ∞, precisely when it is bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In [Ant19] Antieau proves without assumptions on the discrete commutative base ring k, that the HKR-filtration is exhaustive if X is quasi-lci over k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' LΩ1 R/k has Tor-amplitude in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We recover this statement in our situation via the observation that the lci-condition forces � LΩ∗ X/k to be concentrated in degrees (−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, with a result in [Bha12] in the rational setting we can even prove a more drastic result: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If k is a discrete Q-algebra and X a qcqs-scheme over k, then the HKR-filtration is exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the last Corollary we want to prove that � LΩ∗ X/k is bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because X is qcqs and � LΩ∗ −/k is a sheaf, its global sections on X are computed by a finite limit of the value on affines (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, it satisfies to show the claim for X = SpecR with an arbitrary k-algebra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If we write k → R as a filtered colimit of maps (kn → Rn)n∈N in CAlg(D(k0)♥)∆1, where kn is Noetherian and Rn is of finite type over kn, we get � LΩ∗ R/k ≃ colim � LΩ∗ Rn/kn in D(k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Hence, we can further reduce the claim to the case k Noetherian and R finite type over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In this situation the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='10 in [Bha12] gives a concrete description of � LΩ∗ R/k, in particular it sits in homological degree (−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Multiplicative Structure In the Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 the equivalence � Fil• HLΩ∗ R/k ⊗Day Fil• T ktS1�∧ → Fil• T HP(R/k) 10 KONRAD BALS was compatible with the commutative algebra structures on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus we are able to deduce properties of the induced commutative algebra structure on � n∈Z � LΩ∗ R/k[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' But first we will describe algebra structures on these big products more generally: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given a complete and exhaustive commutative multiplicative filtration R• ∈ CAlgFil(Modk) on a commutative algebra R ∈ CAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We define R�t±1� := colim � R• ⊗Day Fil• T ktS1�∧ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If R ∈ CAlgk for an animated commutative ring k, equipped with the constant negatively graded filtration, then we have R�t±1� ≃ RtS1 with respect to the trivial S1-action on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If moreover, π0k is rational, we can even write R((t)) := RtS1 as the unique commutative algebra in Modk with homotopy groups π∗R((t)) for a generator |t| = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 the equivalence refines to a natural equivalence HP(X/k) ≃ � LΩ∗ X/k�t±1� in CAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact, in the situation of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 we can demystify the object � LΩ∗ X/k�t±1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The object R�t±1� does not fully depend on R• as a complete filtered object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will show a very special case, of this feature: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If F • ∈ Modk is a filtered object with F n = 0 for n but finite n, then colim(F • ⊗Day Fil• T ktS1)∧ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, if R• → R• is a map in CAlgFil(Modk)∧ such that the maps induce equivalences for all but finite n, then R�t±1� ≃ R�t±1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' As in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4, we get an equivalence � n∈Z F •−n[−2n] ∼ −→ F •⊗DayFil• T ktS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, the left hand side is already complete: F •−n is complete because it is eventually 0 and the direct sum is in fact a product, because there are only finitely many non-zeros factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, the underlying object of F • is 0 and thus also of the complete filtration F • ⊗Day Fil• T ktS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For the last statement, we note that the construction colim(− ⊗Day Fil• T ktS1)∧ is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 we can have further identifications of commutative algebras HP(X/k) ≃ � LΩ∗ X/k�t±1� ∼ −→ limm LΩ≤m X/k((t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We start in a general setting: Given a complete multiplicative exhaustive filtration R• on a k-algebra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Set R•/Rm to be the filtration with (R•/Rm)(l) := Rl/Rm for l ≤ m and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because R• is complete we have R• ∼ −→ limm R•/Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Checking on associated graded we get an equivalence � R• ⊗Day Fil• T ktS1�∧ ∼ −→ lim m � R•/Rm ⊗Day Fil• T ktS1�∧ and thus the natural map R�t±1� → limm(R/Rm�t±1�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now if R• is eventually constant in negative degrees, and because it is eventually 0 in positive degrees R•/Rm�t±1� ≃ R/Rm((t)) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 and Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, in the concrete situation R• = Fil• H � LΩ∗ X/k, which satisfies this last assumption, we have an easy description of the quotients � LΩ∗ X/k/ � LΩ≥m+1 X/k ≃ LΩ≤m X/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' And now the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 gives an equivalence LΩ≤m X/k�t±1� ≃ � n∈Z LΩ≤m X/k[−2n] on underlying objects, such that the map from � LΩ∗ X/k�t±1� can be identified with the natural map � LΩ∗ X/k → LΩ≤m X/k in each factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular this map is an equivalence in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ We can finally get to the description of the homotopy groups HP∗(X/k) explained in the in- troduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Disregarding the multiplicative structure on HP∗(X/k) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 already gives the PERIODIC CYCLIC HOMOLOGY OVER Q 11 additive identification HP∗(X/k) ∼= � n∈Z π∗+2n � LΩ∗ X/k ∼= �� n∈Z antn : an ∈ π∗+2n � LΩ∗ X/k � with the componentwise addition as stated in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will now show how to describe the multiplication: Given �� n∈Z antn� , �� n∈Z bntn� ∈ HP∗(X/k), then we know that �� n∈Z antn � �� n∈Z bntn � = � n∈Z cntn (4) for some cn ∈ π∗ � LΩ∗ X/k, so that we want to describe these coefficients cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The graded ring π∗ � LΩ∗ X/k can be equipped with the coarsest topology making all maps π∗ � LΩ∗ X/k → π∗LΩ≤m X/k continuous for the discrete topology on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Concretely, this means a neighborhood basis of 0 is given by the kernels of these maps above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, the topology cannot separate points that lie in every single such kernel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' lie in the kernel of the surjective map π∗ � LΩ∗ X/k → lim π∗LΩ≤m X/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In degree i this is precisely given by lim1 πi+1LΩ≤m X/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact lim π∗LΩ≤m X/k is the "Hausdorffization" of this non-Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the equation (4) the coefficient cn is a limit of the net � i+j=n ai · bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It is enough to prove this statement for homogeneous elements, and for simplicity assume that (� n∈Z antn) and (� n∈Z bntn) are both in degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For the general case, one only has to correctly modify the degrees of elements, the arguments are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By definition of the topology on π∗ � LΩ∗ X/k we have to show, that cn − � (i,j)∈Jn ai · bj for finite Jn ⊂ {i + j = n} eventually lies in the kernel of the maps π∗ � LΩ∗ X/k → π∗LΩ≤m X/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5 these maps assemble to ring maps ϕn : HP∗(X/k) → π∗LΩ≤m X/k((t)), where we understand the multiplication of Laurent-series on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, because the coefficients of the target are in degrees ≥ −m as a graded ring, we even know, that ai · bj is sent to 0 in π∗LΩ≤m R/k as soon as i < m/2 or j < m/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' That means for every family of finite sets Jn ⊂ {i + j = n} containing In := {i + j = n : i, j ≥ m/2} ϕn �� n∈Z �� Jn ai · bj � tn � = � n∈Z �� In ϕn(ai) · ϕn(bj) � tn = �� n∈Z ϕn(an)tn � �� n∈Z ϕn(bn)tn � But also by definition we have ϕn �� n∈Z cntn� = �� n∈Z ϕn(an)tn� �� n∈Z ϕn(bn)tn� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, taking the difference and restricting again to single coefficients cn − � Jn ai · bj is sent to 0 in π∗LΩ≤m X/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ This concludes the description of HP∗(X/k) given in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' HP of Schemes In this section, we want to carefully describe the extension of Hochschild and periodic cyclic homology to (derived) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will refer to [Lur18] [Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1], [Lur10] and [Toë14] for an introduction to derived schemes over animated commutative (aka simplicially commutative) rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will only sketch the definition: 12 KONRAD BALS Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For an animated commutative k-algebra R, define the affine derived scheme SpecR to be the pair (|SpecR|, OSpecR) where |SpecR| = |Specπ0R| is a topological space and OSpecR is a CAlgan k/-valued sheaf on |SpecR| with OSpecR(D(f)) ≃ R[f −1] for every elementary open D(f) ⊂ |Specπ0R|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5 A general pair X = (|X|, OX) with |X| a topological space and OX ∈ ShvCAlgan k/(|X|) is called a derived scheme, if there exist an open cover U of X, such that for all U ∈ U we have (U, OX|U) ∼= SpecR6 for some R ∈ CAlgan k/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This notion generalizes ordinary schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular given a derived scheme X, the underlying ringed space π0X := (|X|, π0OX) is an ordinary scheme and we call a derived scheme X affine7, quasi-affine, quasi-compact resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' quasi-separated if π0X is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let X be a derived k-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' A Zariski-sheaf with values in a category C on X is a C-valued sheaf on the topological space |X|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' a functor F : U(X)op → C from the opposite of the poset U(X) of opens of |X|, satisfying F(U) ≃ lim ∅̸=S⊂I finite F(US) for every U = � i∈I Ui ∈ U(X) and with US = Ui0 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' ∩ Uik for S = {i0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' , ik}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given a derived scheme X over k the goal is now to upgrade the functors HH(−/k), HP(−/k): CAlgan k/ → Modk to Zariski-sheaves HHk and HPk on X in order to define HH(X/k) := Γ(X, HHk) and HP(X) := Γ(X, HPk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given a topological space X and Ue a set of open subsets of X, such that 1) Ue forms a basis of the topology of X, 2) Ue is closed under intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Then the adjunction Fun(U(X)op, C) Fun(Uop e , C) res Ran restrict to an equivalence of sheaf cat- egories ShvC(X) ∼ −→ ShvC(Ue) with the induced Grothendieck topology on Ue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If, moreover, Ue consist of quasi-compact opens, then ShvC(X) ≃ Fun′(Uop e , C), where the right hand side consists of those presheaves F : Uop e → C, that satisfy F(∅) = 0 and F(U ∪ V ) ≃ F(U) ×F(U∩V ) F(V ) for U, V, U ∪ V ∈ Ue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The first statement is a special case of the infinity categorical comparison Lemma for Grothen- dieck sites proven in [Hoy14] Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3, and the second claim is [Lur18] Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ We now do the standard procedure of extending an algebraic functor CAlgan k/ → C to a sheaf on geometric objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We proceed in steps: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given a quasi-affine derived scheme X over k, there are Modk-valued sheaves HHk and HPk on X, extending HH(−/k) and HP(−/k), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' for all affine open derived subschemes U ⊂ X, the sheaves recover Hochschild homology, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' periodic cyclic homology: Γ(U, HHk) ≃ HH(OX(U)/k) Γ(U, HPk) ≃ HP(OX(U)/k) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Set Ue to be the set of affine open derived subschemes of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Then HH(−/k) and HP(−/k) give functors Uop e → Modk and let HHk and HPk denote their right Kan extension along Uop e → U(X)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We want to argue, that these are already Zariski-sheaves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because X is quasi-affine, intersections of affines are computed in a surrounding affine derived scheme, and are affine again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The collection Ue, thus, satisfies the conditions 1), 2) of Propos- ition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 and contains only quasi-compact opens, so that we are reduced to checking that the 5The existence of SpecR is deduced in [Lur18] from Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 6Under the appropriate notion of equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 7In fact X is affine, if and only if X = SpecR for R ∈ CAlgan k/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' PERIODIC CYCLIC HOMOLOGY OVER Q 13 functors HH(−/k) and HP(−/k) satisfy the finite limit condition of Fun′(Uop e , Modk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' As the Tate- construction commutes with finite limits, it is enough to only show the claim for Hochschild homo- logy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For R ∈ CAlgan k/ the natural map R → HH(R/k) in CAlgk equips HH(R/k) with a module structure over R, such that for a map of animated commutative rings R → R′ the functoriality induces a map HH(R/k) ⊗R R′ → HH(R′/k) in ModR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now if U ⊂ X is an affine open derived subscheme of X, then for every other affine open V ⊂ U this map HH(OX(U)/k) ⊗OX(U) OX(V ) → HH(OX(V )/k) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Indeed, it suffices to check this locally on V , so we can reduce to distinguished opens D(f) ⊂ V ⊂ U for f ∈ π0OX(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' But using that HH(−/k) commutes with filtered colimits we can identify both sides with HH(OX(U)[f −1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, assume that F : I → Ue is a finite diagram with colimit U as appearing in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4, then by the above HH(OX(F op(−))/k) ≃ HH(OX(U)/k) ⊗OX(U) OX(F op(−)) and we win as tensoring is exact and OX(F op(−)) is a finite limit diagram due to the sheaf condition of OX (using Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 in the other direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given an arbitrary derived k-scheme X, we can furthermore extend Hochschild and periodic cyclic homology to sheaves HHk and HPk on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, for all open qcqs derived subschemes U ⊂ X we have Γ(U, HPk) ≃ Γ(U, HHk)tS1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let Ue now be the set of quasi-affine open derived subschemes of X, which satisfies 1) and 2) of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the last Lemma HH(−/k) and HP(−/k) extend to sheaves on Uop e and by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 thus further extend to sheaves on entire X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now take U ⊂ X a quasi-compact quasi-separated derived open subscheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because of quasi- compactness there exist a finite open cover of U by affine open subschemes U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Un and by the sheaf condition we get Γ(U, HPk) ≃ lim S⊂[1,n] Γ(US, HPk) in the notation of Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Each US is now quasi-affine as an open derived subscheme of an affine and quasi-compact by the quasi-separatedness of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, because the limit above is finite, it satisfies to check the claim for U quasi-compact quasi-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Again, choosing a finite open cover by affines and using that the intersection of affines in quasi-affines is affine again, we can even reduce to the case that U is an affine open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' But in this case Γ(U, HPk) ≃ HP(OX(U)/k) ≃ HH(OX(U)/k)tS1 ≃ Γ(U, HHk)tS1 □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The proof of the last Lemma shows even more: For any sheaf F on a derived scheme X, the sections Γ(U, F) over a qcqs open derived subscheme U are computed as a finite limit of the values of F on affines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Tate Filtration In this section we want to review the construction of the classical Tate-filtration introduced in [GM95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This content is not new and also recently has been explained in [BL22] section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We would like to particular put a focus on multiplicative structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given a representation ρ: S1 → GL(V ) of S1, the representation sphere SV is the one-point compactification of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Furthermore we define SV := Σ∞SV as the suspension spectrum of the representation sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Note that if V is finite dimensional there immediately is an equivalence SV ≃ SdimR V , so that the homotopy type of SV only depends on the dimension of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, the S1-action really uses the representation S1 → GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 14 KONRAD BALS Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For V = C there is the standard representation given by S1 ≃ U(1) ֒→ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Its representation sphere sits in the pushout S1 ∗ ∗ SV with S1-acting freely on itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, after adding basepoints to the top row Σ∞ gives a fibre sequence S[S1] := Σ∞ + S1 → S → SV of spectra with S1-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let V be a finite dimensional representation of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The map 0 → V of repres- entations induces a sequence 0 → V → V ⊕ V → V ⊕ V ⊕ V → · · · which translates to the representation sphere spectra to a Z-graded filtration S•V := · · · → S−2V → S−V → S → SV → S2V → S3V → · · · (5) where S−nV := DSnV is the Spanier-Whitehead dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now if V ̸= 0 all maps have to be non- equivariantly nullhomotopic, but this is definitely not the case with respect to the S1-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will see this later in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given a spectrum X ∈ SpBS1 with S1-action, we define the Tate-filtration FilT XtS1 as · · → � S−2V ⊗X �hS1 → � S−V ⊗X �hS1 →XhS1→ � SV ⊗ X �hS1 → � S2V ⊗X �hS1 → · · · for V the standard representation of S1 constructed in Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This definition would not be sensible if this would not give a filtration on XtS1 and we are bound to prove: Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For X ∈ SpBS1 the Tate filtration Fil• T XtS1 is complete with underlying object XtS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because homotopy fixed points, as a limit, preserve completeness it satisfies to prove that lim S−nV ⊗ X ≃ 0 as a spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' But here we can compute lim S−nV ⊗ X ≃ lim map � SnV , X � ≃ map � colim SnV , X � ≃ 0 because the colimit goes along nullhomotopic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, as already indicated, those maps are not equivariantly nullhomotopic In fact every map S−nV ⊗ X → S(−n+1)V ⊗ X induces an equivalence on the S1-Tate construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Indeed, via Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 we can identify the fibre as S−nV ⊗ S[S1], which is an induced S1-spectrum, such that Tate vanishes on this fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now we can look at the Z-indexed fibre sequences defining the Tate constructions of S−nV ⊗ X: Σ � S−nV ⊗ X � hS1 � S−nV ⊗ X �hS1 � �� � ≃Filn T XtS1 � S−nV ⊗ X �tS1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the observation above the right hand filtration is constant at XtS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The colimit of the left hand filtration vanishes, because commuting the colimit with Σ(− ⊗ X)hS1 reduces again to computing a filtered colimit along nullhomotopic maps, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus together we see colim Filn T XtS1 ≃ XtS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ PERIODIC CYCLIC HOMOLOGY OVER Q 15 We are interested in possible algebra structures on the Tate filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because (−)tS1 is lax monoidal, XtS1 for an algebra X ∈ Alg(Sp) inherits an algebra structure again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, the question of algebra structures on FilT XtS1 with respect to the Day convolution is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will use the following different description of the filtered category as a modules over a graded algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This insight comes from Lurie in [Lur15] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 and in this form is in [Rak20] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a stable symmetric monoidal category C with unit 1, let 1[β] denote the underlying graded object of the unit in Fil(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It is a commutative algebra in Gr(C) with underlying graded object � n≤0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Every object in the symmetric monoidal category Fil(C) is canonically a module over the unit, such that the symmetric monoidal forgetful functor Fil(C) → Gr(C) refines to a functor Fil(C) → Mod 1[β](Gr(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact remembering this action of 1[β] recovers the full filtered object: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8 ([Rak20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a symmetric monoidal stable category C the above functor Fil(C) → Mod 1[β](Gr(C)) is an equivalence of symmetric monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In our situation we want to use this for C = SpBS1 Q and show that the filtered object S−•V ⊗ Q is a commutative algebra in Fil(SpBS1 Q ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' There is also an algebraic description of this category due to Greenlees-Shipley [GS09] in the non-Borel-complete setting and later as we use it here by [MNN17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Similar to above, because again in SpQ every object carries a canonical module structure over the unit Q, the lax functor (−)hS1 : SpBS1 Q → SpQ refines to a functor into ModQhS1 (SpQ) and we have as a special case of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='35 in [MNN17]: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='9 ([MNN17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The functor (−)hS1 : SpBS1 Q → ModQhS1 (SpQ) is fully faithful with essential image given by those modules over QhS1 ≃ Q�t� that are complete with respect to the t-adic filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The Thom isomorphism over Q for complex vector bundles over BS1 gives an S1-equivariant equivalence of SV ⊗Q ≃ Q[2] with trivial S1-action on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus the map S⊗Q → SV ⊗Q ≃ Q[2] in SpBS1 Q corresponds to an Q�t�-module map Q�t� → Q�t�[2] for |t| = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular as a Q�t�- module map it is determined by the image of 1 in Q·t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because this map is not 0 as seen in the proof of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6, up to a unit, it is given by multiplication by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' More generally this argument gives an identification of the image of the filtration S−•V ⊗ Q under (−)hS1 with the filtration · · t−→ Q�t�[−2n] t−→ Q�t� t−→ Q�t�[2n] t−→ · · · of Q�t�-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The filtration S−•V ⊗Q can be given a commutative algebra structure in Fil(SpBS1 Q ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Under the symmetric monoidal equivalences from the cited Theorems B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='9 we are reduced to equip the underlying graded object � n∈Z Q�t�[−2n] of (S−•V ⊗Q)hS1 with a commutative algebra structure over Q�t�[β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' To avoid confusion, let us introduce a formal variable s in grading degree −1 and homological degree 2 to get an identification of underlying objects � n∈Z Q�t�[−2n] ≃ Q�t�[s±1], which is the free graded commutative Q�t�-algebra on the variables s±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular sending β to s · t gives Q�t�[s±1] the desired commutative algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a commutative ring k with Q ⊂ π0k and R ∈ CAlgBS1 k the filtration FilT RtS1 permits the structure of a commutative algebra in Fil(Modk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 16 KONRAD BALS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By construction FilT (−)tS1 is the composite ModBS1 k (−)⊗(S−•V ⊗k) −−−−−−−−−−→ Fil(ModBS1 k ) (−)hS1 −−−−→ Fil(Modk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The second functor has a canonical lax structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because k is a commutative algebra over Q, also (S−•V ⊗ k) inherits a commutative algebra structure via Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='10 and thus FilT (−)tS1 refines to a lax symmetric monoidal functor and sends commutative algebras in ModBS1 k to commutative algebras in Fil(Modk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Given C∗ ∈ CDGAQ ι֒−→ SpBS1 Q we would like to conclude this section with the comparison of the induced filtration on Fil≥0 T UCtS1 ∗ on the zeroth part Fil0 T UCtS1 ∗ ≃ UChS1 ∗ to concrete filtrations on the chain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the notation of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11, there is an identification of the t-adic filtration on UChS1 ∗ ≃ (UC∗�t�, td) with the Tate filtration Fil≥0 T UCtS1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Using the cocommutative bialgebra A := Q[ǫ]/ǫ2 as defined in Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1 we have the symmetric monoidal equivalence of categories SpBS1 Q ∼ −→ ModASpQ (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Therefore the filtration Fil≥0 T UCtS1 ∗ reads as · · → mapA(Q, Q[−4] ⊗ C∗) → mapA(Q, Q[−2] ⊗ C∗) → mapA(Q, Q ⊗ C∗) ≃ UChS1 ∗ By duality this filtration is equivalently induced by the maps Q[2n] → Q[2n + 1] from S−•V ⊗ Q for n ≥ 0 in the first variable of the mapping spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Choosing P∗ = (A⟨t∨⟩, dP ) for t∨ primitive in degree 2 and dP (t∨) = ǫ as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11, these maps P∗[2n] · · k · (t∨)2 k · ǫt∨ k · t∨ k · ǫ k P∗[2n + 1] · · k · t∨ k · ǫ k ∼ 0 ∼ 0 ∼ 0 are uniquely determined as A-module maps by the image of t∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because again the map is non-zero, the image of t∨ has to be a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, up to isomorphism the maps ChS1 ∗ [−2n − 2] → ChS1 ∗ [−2n] are given by multiplication with the dual t in (C∗�t�, td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ References [Ada56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Adams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' ‘On The Cobar Construction’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In: Proceedings of the National Academy of Sciences 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7 (1956), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 409–412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [AMN18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Antieau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Mathew and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Nikolaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' ‘On the Blumberg–Mandell Künneth theorem for TP’.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' ‘On the (co)homology of commutative rings’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In: Applications of Categorical Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [Rak20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Raksit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Hochschild homology and the derived de Rham complex revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' arXiv: 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='02576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [Toë14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Toën.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' ‘Derived algebraic geometry’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In: EMS Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Math Schi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1 2 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 153– 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Weibel and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Geller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' ‘Étale descent for hochschild and cyclic homology’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In: Commentarii Mathematici Helvetici 66 (1991), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 368–388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' WWU Münster, Mathematisches Institut, Einsteinstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 62, 48149 Münster, Germany Email address: konrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='bals@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'}