Equivariant algebraic vs. topological K-homology Atiyah-Segal-style
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 the site whose objects are locally separated algebraic spaces U with \(U\to X\) finitely presented, étale and G- isovariant. A morphism is a G-equivariant map \(U\to V\) compatible with the maps to X. Such a morphism is étale and isovariant. The topos ``X/G'' is well adapted to geometric questions. On the other hand one can define another topos ``X//G'', better adapted to topos-theoretic questions, like finiteness of cohomological dimension and pasting. ``X/G'' and ``X//G'' are shown to be equivalent topoi. Part of a fundamental result is the following descent theorem, which says that under certain conditions on S and the prime l, there is a homotopy equivalence \[ G/l^{\nu}(G,X)[\beta^{-1}]\simeq {\mathbb{H}}^.(``X/G'';\quad G/l^{\nu}(G,p^{-1}(\quad))[\beta^{-1}]), \] where \(G(G,p^{-1}(\quad))\) is the presheaf of spectra on the site of ``X/G'', sending an isovariant étale \(U\to X\) to \(G(G,p^{- 1}(U/G))=G(G,U)\). \({\mathbb{H}}^.\) denotes the hypercohomology spectrum. For X of finite Krull dimension a Zariski version also exists without the need of \(\beta\)-inversion and with ``X/G'' replaced by \(``X/G_{Zar}''\), the topos of the site of G-invariant open subspaces of X. Translation in terms of homotopy groups (with \(F_ n\) short for \(\pi_ nF\) for a spectrum F...) gives a strongly converging Fary spectral sequence \[ E_ 2^{p,q}=H^ p(``X/G'';\quad \tilde G/l_ q^{\nu}(G,p^{- 1}(\quad))[\beta^{-1}])\Rightarrow G/l^{\nu}_{q-p}(G,X)[\beta^{- 1}]. \] For X a scheme of finite type over an algebraically closed field k of characteristic 0, acted on by a smooth linear algebraic group G over k, one obtains a proof of the fact that \(G/l^{\nu}(G,X)[\beta^{-1}]\) satisfies the Kazhdan-Lusztig axioms for an equivariant K-homology theory suitable for p-adic representation theory. Next a proof is given of Segal's Concentration Theorem, which says that localization at a prime \(\rho\) of the representation ring R(G) of a linear algebraic group G over an algebraically closed field k of characteristic 0 induces a homotopy equivalence \[ G/l^{\nu}(G,X^{(\rho)})[\beta^{-1}]_{(\rho)}\simeq G/l^{\nu}(G,X)[\beta^{-1}]_{(\rho)}, \] where \(X^{(\rho)}\to X\) is a well defined equivariant immersion. Such a prime \(\rho\) corresponds to a conjugacy class of a diagonalizable subgroup of G. The last paragraph contains comparison results for a linear algebraic group G over \({\mathbb{C}}\), with a maximal compact subgroup M of G(\({\mathbb{C}})\). Under suitable conditions there are homotopy equivalences with topological K- homology \[ G/l^{\nu}(G,X)[\beta^{-1}]\simeq G/l^{\nu,Atiyah- Segal}(M,X({\mathbb{C}})) \] and with topological K-cohomology \[ K/l^{\nu}(G,X)[\beta^{-1}]\simeq K/l^{\nu,Atiyah- Segal}(M,X({\mathbb{C}})), \] where \(K/l^{\nu,Atiyah-Segal}(M,X({\mathbb{C}}))\) denotes the mod \(l^{\nu}\) reduction of the spectrum of M-equivariant complex topological vector bundles on X(\({\mathbb{C}})\). Similarly for G. Actually one has to be careful with the notion of equivariant homotopy equivalence. In the above the equivalences are with the ``Bredon-type'' equivariant topological K-(co)homologies.
group scheme over a base scheme acting on a noetherian locally separated algebraic space
Kazhdan-Lusztig axioms for an equivariant K-homology theory
scientific article; zbMATH DE number 4069820
scientific article