The cyclic homology of an exact category

McCarthy, Randy

doi:10.1016/0022-4049(94)90091-4

Cited by 75 publications

(89 citation statements)

References 18 publications

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Section: The Trace Mapsmentioning

confidence: 99%

“…The lift ntr 0 of this map is explicitly described on page 286 in [28]. The remaining horizontal maps are just the inclusions of the zero simplices.…”

Section: The Trace Mapsmentioning

confidence: 99%

“…The trace maps for additive categories. We now review the construction of K-theory for additive categories, and of the map h and the trace maps ntr and dtr for k-linear categories with finite sums, following the ideas of [28], [11] and [9].The following commutative diagram is natural in the k-linear category A : Here, the lower horizontal map dtr 0 is given by sending an object to the corresponding identity morphism. The lift ntr 0 of this map is explicitly described on page 286 in [28].…”

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Detecting $K$-Theory by Cyclic Homology

Lück

Reich

2006

Proc. Lond. Math. Soc.

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Abstract. We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology. Key words: algebraic K-theory of group rings, Hochschild homology, cyclic homology, trace maps. Mathematics Subject Classification 2000: 19D55. Dedicated to memory of Michel Matthey. Introduction and statement of resultsFix a commutative ring k, referred to as the ground ring. Let R be a k-algebra, i.e. an associative ring R together with a unital ring homomorphism from k to the center of R. We denote by HH ⊗ k * (R) the Hochschild homology of R relative to the ground ring k, and similarly by HC ⊗ k * (R), HP ⊗ k * (R) and HN ⊗ k * (R) the cyclic, the periodic cyclic and the negative cyclic homology of R relative to k. Hochschild homology receives a map from the algebraic K-theory, which is known as the Dennis trace map. There are variants of the Dennis trace taking values in cyclic, periodic cyclic and negative cyclic homology (sometimes called Chern characters), as displayed in the following commutative diagram. For the definition of these maps, see [18, Chapters 8 and 11] and Section 5 below. In the following we will focus on the case of group rings RG, where G is a group and we refer to the k-algebra R as the coefficient ring. We investigate the following question.Question 0.2. Which part of K * (RG) ⊗ Z Q can be detected using linear trace invariants like the Dennis trace to Hochschild homology, or its variants with values in cyclic homology, periodic cyclic homology and negative cyclic homology ?For any group G, we prove "detection results", which state that certain parts of K * (RG) ⊗ Z Q can be detected by the trace maps in diagram 0.1, accompanied by "vanishing results", which state that a complement of the part which is then known to be detected is mapped to zero. For the detection results, we only make assumptions on the coefficient ring R, whereas for the vanishing results we additionally need the Farrell-Jones Conjecture for RG as an input, compare Example 1.2. Modulo the Farrell-Jones Conjecture, we will give a complete answer to Question 0.2 for instance in the case of Hochschild and cyclic homology, when the coefficient ring R is an algebraic number field F or its ring of integers O F . We will also give partial results for periodic cyclic and negative cyclic homology. All detection results are obtained by using only the Dennis trace with values in HH ⊗ k * (RG), whereas all vanishing results hold even for the trace with values in HN ⊗ Z * (RG), which, in view of diagram (0.1), can be viewed as the best among the considered trace invariants. (Note that for a k-algebra R every homomorphism k ′ → k of commutative rings leads to a homomorphism HN ⊗ k ′ * (R) → HN ⊗ k * (R). Similar for Hochschild, cyclic and periodic cyclic homology.) We have no example where the extra effort that goes into the construction of the variants with values in cyclic, periodic cyclic or negative cyclic homology yields mo...

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Section: The Trace Mapsmentioning

confidence: 99%

“…The lift ntr 0 of this map is explicitly described on page 286 in [28]. The remaining horizontal maps are just the inclusions of the zero simplices.…”

Section: The Trace Mapsmentioning

confidence: 99%

mentioning

confidence: 99%

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Detecting $K$-Theory by Cyclic Homology

Lück

Reich

2006

Proc. Lond. Math. Soc.

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“…The category D b R-c (k M ) of constructible sheaves on a compact real analytic manifold M is "proper" in the sense of Kontsevich (that is, Ext finite) but it does not admit a Serre functor (in the sense of Bondal-Kapranov) and it is not clear whether it is smooth (again in the sense of Kontsevich). However this category naturally appears in Mirror Symmetry (see [FLTZ10]) and it would be a natural question to try to understand its Hochschild homology in the sense of [McC94,Ke99]. We don't know how to compute it, but the above construction, with the use of µhom(k ∆ M , ω ∆ M ), provides an alternative approach of the Hochschild homology of this category.…”

Section: Introductionmentioning

confidence: 99%

Microlocal Euler classes and Hochschild homology

Kashiwara

Schapira

2013

J. Inst. Math. Jussieu

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We define the notion of a trace kernel on a manifold M . Roughly speaking, it is a sheaf on M ×M for which the formalism of Hochschild homology applies. We associate a microlocal Euler class to such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle T * M over M and we prove that this class is functorial with respect to the composition of kernels.This generalizes, unifies and simplifies various results of (relative) index theorems for constructible sheaves, D-modules and elliptic pairs.

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“…Connes's construction made use of idempotents, a generalized trace map, etc. Later, in the nineties, McCarthy [7] and Keller [5] extended 1 the Chern character maps (1.1) from k-algebras to dg categories; see §2 for the notion of a dg category. Given a commutative and unital base ring k, we have the natural transformations…”

Section: Introductionmentioning

confidence: 99%