Higher algebraic \(K\)-theory for actions of diagonalizable groups. (English) Zbl 1032.19001
Invent. Math. 153, No. 1, 1-44 (2003); erratum ibid. 161, No. 1, 219-224 (2005).
This paper gives a decomposition theorem for the higher algebraic \(K\)-theory, \(K_* (X,G)\), of a separated noetherian regular algebraic space, \(X\), over a scheme, \(S\), acted on by a diagonalizable group scheme, \(G\), of finite type over \(S\). In an earlier paper [G. Vezzosi and A. Vistoli, Duke Math. J. 113, 1-55 (2002; Zbl 1012.19002)] the authors showed that if \(G\) acts with finite stabilizers then \(K_* (X,G)\) can be expressed as a product after inverting some primes. A similar result holds if the stabilizers are of constant dimension. The present paper considers the general case, in which the product decomposition must be replaced by a fibred product. The main theorem is that if \(X_s\) denotes the locus of \(X\) where the stabilizer is of dimension \(s\), \(N_s\) denotes the normal bundle of \(X_s\) in \(X\) and \(N_{s, s-1}\) the subspace where the stabilizer has dimension \(s-1\) then the restriction homomorphisms \(K_*(X,G) \to K_* (X_s, G)\) induce an isomorphism
\[
K_*(X,G) \simeq K_* (X_n, G) \times_{K_* (N_{n,n-1}, G)} K_* (X_{n-1}, G) \times _{K_* (N_{n-1, n-2}, G)}\times\ldots\times_{K_* (N_{1,0},G)} K_* (X_0, G)
\]
where \(n\) is the dimension of \(G\). Using this the authors prove a version of Brion’s theorem 3.4 [M. Brion, Transform. Groups 2, 225-267 (1997; Zbl 0916.14003)] for algebraic \(K\)-theory. They also analyze the case in which \(X\) is a smooth toric variety.
Reviewer: Keith Johnson (Halifax)
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