Higher \(K\)-theory of toric stacks (Q2792158)

The authors develop several tools for computing the \(K\)-theory of quotient stacks and apply them, in particular, to toric and spherical varieties. They thereby recover and generalize results of \textit{R. Morelli} [Adv. Math. 100, No. 2, 154--182 (1993; Zbl 0805.14025)], \textit{Y. Takeda} [Osaka J. Math. 35, No. 1, 73--81 (1998; Zbl 0918.19002)], \textit{L. A. Borisov} and \textit{R. P. Horja} [Contemp. Math. 401, 21--42 (2006; Zbl 1171.14301)], \textit{G. Vezzosi} and \textit{A. Vistoli} [Invent. Math. 153, No. 1, 1--44 (2003; Zbl 1032.19001); erratum ibid. 161, No. 1, 219--224 (2005)], and others.NEWLINENEWLINEOne of the key results is the existence of a strongly convergent Eilenberg-Moore type spectral sequence for equivariant \(K\)-theory, as follows. Let \(G\) be a connected split reductive group with torsion-free \(\pi_1(G)\), let \(T\subset G\) be a maximal torus and let \(S\to T\) be a diagonalizable group scheme mapping to \(T\). For any smooth scheme \(X\) on which \(G\) acts, the authors construct a spectral sequence computing the \(S\)-equivariant \(K\)-theory of \(X\) from its \(G\)-equivariant \(K\)-theory (Cor.~4.3): NEWLINE\[NEWLINE \mathrm{Tor}_*^{RG}(RS, K_*^G(X)) \Rightarrow K^S_*(X) NEWLINE\]NEWLINE This generalizes the spectral sequence of \textit{A. S. Merkur'ev} [St. Petersbg. Math. J. 9, No. 4, 815--850 (1998); translation from Algebra Anal. 9, No. 4, 175--214 (1997; Zbl 0897.19004)], where \(S\) was trivial. There are no assumptions on the homomorphism \(S\to T\). When \(G=T\), the spectral sequence degenerates for any smooth projective toric variety \(X\) containing \(T\) as dense torus (Thm~1.1). This is the case the authors are mainly interested in.NEWLINENEWLINEUsing similar techniques, the authors show that the higher \(S\)-equivariant \(K\)-theory of any smooth projective toric variety as above is related to its \(S\)-equivariant \(K_0\) in the simplest possibly way (Thm~1.2): NEWLINE\[NEWLINE K_0^S(X)\otimes_{RS} K^S_*(k) \cong K^S_*(X) NEWLINE\]NEWLINE More generally, this is true for any smooth projective \(T\)-linear scheme. The class of \(T\)-linear schemes is introduced by the authors as an equivariant generalization of the concept of linear schemes of \textit{B. Totaro} [Forum Math. Sigma 2, Article ID e17, 25 p. (2014; Zbl 1329.14018)]. Besides toric varieties (Prop.~3.6), it includes, in particular, all spherical varieties in characteristic zero (Cor.~3.7).NEWLINENEWLINEThe results are applied to compute the \(K\)-theory of weighted projective spaces (Thm~5.6), to establish an equivariant Leray-Hirsch theorem (Thm~6.3), and to compute the (higher) \(K\)-theory of toric bundles (Thm~1.3).NEWLINENEWLINEThe main results apply more generally to non-smooth schemes if \(K\)-theory is replaced by \(G\)-theory. The language of toric stacks developed in [\textit{A. Geraschenko} and \textit{M. Satriano}, Trans. Am. Math. Soc. 367, No. 2, 1033--1071 (2015; Zbl 1346.14034)] is used throughout.