RiemannRoch for Algebraic Stacks: I
Author: Joshua⋆, R.
Source: Compositio Mathematica, Volume 136, Number 2, April 2003 , pp. 117-169(53)
In this paper we establish RiemannRoch and LefschtezRiemannRoch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The RiemannRoch theorem is established as a natural transformation between the G-theory of algebraic stacks and topological G-theory for stacks: we define the latter as the localization of G-theory by topological K-homology. The LefschtezRiemannRoch is an extension of this including the action of a torus for DeligneMumford stacks. This generalizes the corresponding RiemannRoch theorem (LefschetzRiemannRoch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G-theory (as well as rational G-theory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological G-theory on schemes as proved by Thomason.
Document Type: Research Article
Affiliations: Department of Mathematics, Ohio State University, Columbus, OH, 43210, U.S.A. e-mail: firstname.lastname@example.org
Publication date: April 1, 2003