Additive \(K\)-theory and crystalline cohomology
type: Item
instance of: scholarly article
title: Additive \(K\)-theory and crystalline cohomology
zbMATH Open document ID: 0585.18007
DOI: 10.1007/BF01078391
author: B. L. Tsygan
author: B. Feigin
publication date: 1985-01-01T00:00:00Z
review text: In recent years many new (co)homology theories of (non)commutative (operator) algebras have been discovered: The homological \(K\)-functor Ext of \textit{L. G. Brown}, \textit{R. G. Douglas} and \textit{P. A. Fillmore} [see Bull. Am. Math. Soc. 79, 973--978 (1973; Zbl 0277.46052)] having after all some interesting application in the structure theory of \(C^*\)-algebras [see the reviewer, Funkts. Anal. Prilozh. 9, No. 1, 63--64 (1975; Zbl 0347.46068)], Kasparov's \(KK\)-functor [see \textit{G. G. Kasparov}, Izv. Akad. Nauk SSSR, Ser. Mat. 44, 571--636 (1980; Zbl 0448.46051)], Connes' cyclic homology \(HC\)-functor [see \textit{M. Karoubi}, C. R. Acad. Sci., Paris, Sér. I 297, 381--384, 447--450, 513--516, 557--560 (1983; Zbl 0528.18008, Zbl 0528.18009, Zbl 0528.18010, Zbl 0532.18009)], and the Gel'fand-Fuks cohomology theory of infinite-dimensional Lie algebras. The most important point in these developments is replacing the structural function sheaf by an arbitrary (non)commutative algebra. The paper under review is a new account of this fact, where the algebra of functions on a manifold is replaced by a (non)commutative algebra. One considers for each algebra \(A\) over a field \(K\) of characteristic zero the standard bimodule resolution \[ R_* :\quad... \to A\otimes A\otimes A\to^{\delta}A\otimes A\to^{\delta}A\to 0 \] with the differential \(\delta (a_ 1\otimes...\otimes a_ k)=\sum^{k-1}_{i=1}(-1)^ ia_ i\otimes...\otimes a_ ia_{i+1}\otimes...ca_ k\), the standard complex \(C_*(A)\) of the Lie algebra \(\mathfrak{gl}_{\infty}(A)\) of infinite matrices of finite type over \(A\) with coefficients in the unit module, and finally a complex homomorphism \[ \phi: C_*(A)\to (R_*,\delta),\quad \phi ((m_ 1\otimes a_ 1)\wedge...\wedge (m_ k\otimes a_ k))=\sum_{i}\text{sgn}(i)\text{tr}(m_{i_ 1}...m_{i_ k}) a_{i_ 1}\otimes...\otimes a_{i_ k}. \] The image of \(\phi\) is the subcomplex \(\bar R_*\subset R_*\) consisting of chains that are invariant under the transfer map \(t\…
Mathematics Subject Classification ID: 19K99
Mathematics Subject Classification ID: 19K33
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 46M15
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 22E41
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 55R45
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 57R32
zbMATH DE Number: 3937380
zbMATH Keywords: cohomology theories of noncommutative operator algebras
zbMATH Keywords: Lie
zbMATH Keywords: algebra of infinite matrices of finite type
zbMATH Keywords: homological K-functor
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: Kasparov's KK-functor
zbMATH Keywords: cyclic homology
zbMATH Keywords: Gel'fand-Fuks cohomology theory of infinite-dimensional Lie
zbMATH Keywords: algebras
zbMATH Keywords: additive K-functor
zbMATH Keywords: derived functors
zbMATH Keywords: Chern characters
zbMATH Keywords: Bott periodicity
zbMATH Keywords: crystalline cohomology
zbMATH Keywords: differential graded algebra
zbMATH Keywords: de Rham complex
zbMATH Keywords: Gel'fand-Fuks cohomology theory of infinite-dimensional Lie algebras
reviewed by: Do Ngoc Diep
MaRDI profile type: Publication
cites work: Q4115296
cites work: Homologie nicht-additiver Funktoren. Anwendungen
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cites work: Q3667036
cites work: Q3310035
cites work: Q3314036
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full work available at URL: https://doi.org/10.1007/bf01078391
OpenAlex ID: W2004334856
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3937380