Equivariant algebraic vs. topological K-homology Atiyah-Segal-style
type: Item
instance of: scholarly article
title: Equivariant algebraic vs. topological K-homology Atiyah-Segal-style
zbMATH Open document ID: 0655.55002
DOI: 10.1215/S0012-7094-88-05624-4
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let G be a group scheme over a suitable base scheme S, acting on a noetherian locally separated algebraic space X. \(G\to S\) is supposed to be flat, separated and of finite type. Then one can associate to the exact category of coherent sheaves of \({\mathcal O}_ X\)-modules with compatible G-action and to the exact subcategory of algebraic G-vector bundles, the K-theory spectra G(G,X) and K(G,X), respectively. These (topological) spectra admit multiplication by a prime power \(l^{\nu}\), such that one has a homotopy fibre sequence \(G\to^{l^{\nu}}G\to G/l^{\nu}\) (similarly for K). \(G/l^{\nu}=G/l^{\nu}(G,X)\) is called the mod \(l^{\nu} reduction\) of G(G,X). It is known that this algebraic \(G/l^{\nu}\) bears some relation to the topological \(G/l^{\nu,Top}\). Analogous facts hold for K. Topological K-theory goes equipped with Bott periodicity. Algebraic K-theory has no such property. To overcome this, one imposes Bott periodicity by inverting the so-called Bott element \(\beta\) to obtain spectra \(G/l^{\nu}(G,X)[\beta^{-1}]\) and \(K/l^{\nu}(G,X)[\beta^{-1}]\). These are the objects of the present paper. First, the basic properties of G(G,X) and K(G,X) are recalled: functorial behaviour, localization exact sequences, projective space bundle theorem, homotopy property and Poincaré duality. For \(H\subset G\) a closed subgroup acting on \(G\times X\) by \(h(g,x):=(gh^{-1},hx)\), one denotes \(G\times X\) modulo this action by \(G\times^ HX\) and one obtains Morita equivalence \(G(G,G\times^ HX)\simeq G(H,H\times^ HX)\simeq G(H,X)\) and in case G is reductive with a maximal torus T of a Borel subgroup of G, there is a naturally split monomorphism \(G(H,X)\to G(T,G\times^ HX)\). Next, for a group scheme G acting on two algebraic spaces X and Y, one calls a map \(f: X\to Y\) isovariant if f is equivariant and if, moreover, it induces an isomorphism of isotopy groups \(G_ X\simeq G_ Y\simeq_ YX\). Then the orbit topos ``X/G'' is defined as the Grothendieck topos of t…
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 4069820
zbMATH Keywords: group scheme over a base scheme acting on a noetherian locally separated algebraic space
zbMATH Keywords: algebraic G-vector bundles
zbMATH Keywords: K-theory spectra
zbMATH Keywords: Topological K-theory
zbMATH Keywords: Bott periodicity
zbMATH Keywords: Algebraic K-theory
zbMATH Keywords: Bott element
zbMATH Keywords: descent theorem
zbMATH Keywords: Fary spectral sequence
zbMATH Keywords: Kazhdan-Lusztig axioms for an equivariant K-homology theory
zbMATH Keywords: Segal's Concentration Theorem
zbMATH Keywords: localization
author: Robert W. Thomason
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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