Decomposition theorems for the K-theory of algebraic stacks

A. Vistoli proved a decomposition theorem for the rational equivariant algebraic K-theory of a variety under the action of a finite group G. Subsequently, G. Vezzosi and A. Vistoli proved a decomposition theorem for the equivariant K-theory of a noetherian scheme, in the more general case where G is any affine algebraic group (with some mild hypothesis about the action). Moreover, B. Toen defined a Riemann-Roch map from the rational algebraic K-theory of a tame Deligne-Mumford quotient stack to the étale K-theory of its inertia. He proved that this map is an isomorphism and that it is covariant with respect to proper maps.In this thesis we first give a geometric definition of the Vezzosi-Vistoli decomposition, interpreting the pieces as corresponding to the components of the cyclotomic inertia. When the map from the cyclotomic inertia to the stack is finite, we can define a Riemann-Roch map in Toen's style. We prove that this map is an isomorphism and it is covariant with respect to proper relatively tame maps; moreover in some favourable circumstances we explicitly compute its inverse map, and show that we can recover Toen's one when the stack is tame Deligne-Mumford. Then we generalize Vistoli's result to more general algebraic (co)homology theories having the Mackey property and admitting localization long exact sequences. In general, the pieces are indexed by conjugacy classes of subgroups of G. Our construction is based on some result about a decomposition of the rational Burnside ring of a finite group, which stands behind the classical splitting theorems for equivariant spectra in stable equivariant homotopy theory. Applying this result to the case of Borne's modular K-theory we exhibit a case where the splitting is indexed by not necessarily abelian subgroups.