Étale descent for hochschild and cyclic homology

Weibel, Charles A.; Geller, Susan C.

doi:10.1007/bf02566656

Cited by 59 publications

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“…For an affine scheme X = Spec A, HH n (X) agrees with the classical group HH n (A) by [WG,4.1]. Note that X can have negative Hochschild homology modules (viz.…”

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confidence: 99%

“…The key axiom (0.2) for HH is a consequence ofétale descent. Hochschild and cyclic homology of schemes are related by a natural "SBI" sequence [WG,4.5 Since hypercohomology of unbounded complexes is an ambiguous notion, I have included an appendix to show how the "Cartan-Eilenberg" hypercohomology used in this paper (and in [CE] [T], [W], [WG], etc.) compares with hypercohomology in the sense of derived categories.…”

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confidence: 99%

“…However, when X is quasi-compact and quasi-separated, the argument of 0.5 shows that only finitely many negative Hochschild modules can be nonzero. For example, HH n (X) = 0 for n < − dim(X) whenever X is finite-dimensional and noetherian over k [WG,4.3].…”

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confidence: 99%

“…are not quasi-coherent for n > 0, the following proposition, which is a paraphrase of part of [WG,0.4], states that the chain complex C h * has quasi-coherent homology. To formulate it, let HH n denote the n th sheaf homology of C h * .…”

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confidence: 99%

“…Hence HH n is also the sheafification of the presheaf U → HH n (O X (U )). Proposition 1.2 [WG,0.4]. Let X be a scheme over k. Then each HH n is a quasi-coherent sheaf on X.…”

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confidence: 99%

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Cyclic homology for schemes

Weibel

1996

Proc. Amer. Math. Soc.

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Abstract. Using hypercohomology, we can extend cyclic homology from algebras to all schemes over a ring k. By 'extend' we mean that the usual cyclic homology of any commutative algebra agrees with the cyclic homology of its corresponding affine scheme.The purpose of this paper is to show that there is a cyclic homology theory HC * of schemes over a commutative ring k, extending the usual cyclic homology HC * of k-algebras. By a cyclic homology theory for schemes over k we mean a family of graded k-modules HC n (X) associated to every scheme X over k which satisfy:(0.1) they are natural and contravariant in X; (0.2) for each affine scheme X = Spec A, there are natural isomorphismsWe discuss uniqueness of a cyclic homology theory briefly in Remark 0.5 below. We have chosen homological indexing because of axiom (0.2), and because cohomological indexing (HC n = HC −n ) would concentrate the nonzero groups in negative degrees.The construction is simple: just sheafify Connes' (b, B) double chain complex and take Cartan-Eilenberg hypercohomology (as defined in the appendix). Axioms (0.1) and (0.3) are immediate from generalities about hypercohomology of unbounded complexes. The point of this paper is that axiom (0.2) also holds. In the spirit of Grothendieck's seminal letter [GdR], Geller and I also proved in [WG] that sheafifying and taking hypercohomology yields a "Hochschild homology

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“…For an affine scheme X = Spec A, HH n (X) agrees with the classical group HH n (A) by [WG,4.1]. Note that X can have negative Hochschild homology modules (viz.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Hence HH n is also the sheafification of the presheaf U → HH n (O X (U )). Proposition 1.2 [WG,0.4]. Let X be a scheme over k. Then each HH n is a quasi-coherent sheaf on X.…”

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confidence: 99%

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Cyclic homology for schemes

Weibel

1996

Proc. Amer. Math. Soc.

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A Stronger Derived Torelli Theorem for K3 Surfaces

Lieblich

Olsson

2017

Geometry Over Nonclosed Fields

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In an earlier paper the notion of a filtered derived equivalence was introduced, and it was shown that if two K3 surfaces admit such an equivalence then they are isomorphic. In this paper we study more refined aspects of filtered derived equivalences related to the action on the cohomological realizations of the Mukai motive. It is shown that if a filtered derived equivalence between K3 surfaces also preserves ample cones then one can find an isomorphism that induces the same map as the equivalence on the cohomological realizations.

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Hochschild Homology

Loday

1998

Grundlehren Der Mathematischen Wissenschaften

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Étale descent for hochschild and cyclic homology

Cited by 59 publications

References 13 publications

Cyclic homology for schemes

Cyclic homology for schemes

A Stronger Derived Torelli Theorem for K3 Surfaces

Hochschild Homology

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