G-theory of root stacks and equivariant K-theory

Abstract

Using the description of the category of quasi-coherent sheaves on a root stack, we compute the G-theory of root stacks via localisation methods. We apply our results to the study of equivariant K-theory of algebraic varieties under certain conditions. Proof. This follows by combining Corollaries 3.10 and 3.15. So we have:G(X L, r ) ∼ = K(EP(X, L, r)), and we reduced the problem to describing the K-theory of the (abelian) category of coherent extendable pairs EP(X, L, r).We are going to use several splittings of the category of coherent extendable pairs to simplify the latter K-theory. The first step is this Lemma 3.29. If X is a locally noetherian scheme, then in notation of section 3.3 one has:

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