On the K-theory of tame Artim stacks

This thesis pertains to the algebraic K-theory of tame Artin stacks. Building on earlier work of Vezzosi and Vistoli in equivariant K-theory, which we translate in stacky language, we give a description of the algebraic K-groups of tame quotient stacks. Using a strategy of Vistoli, we recover Grothendieck-Riemann-Roch-like formulae for tame quotient stacks that refine Toën’s Grothendieck-Riemann-Roch formula for Deligne-Mumford stacks (as it was realized that the latter pertains to quotient stacks since it relies on the resolution property). Our formulae differ from Toën’s in that, instead of using the standard inertia stack, we use the cyclotomic inertia stack introduced by Abramovich, Graber and Vistoli in the early 2000s. Our results involve the rational part of the K'-theory of the object considered. We establish a few conjectures, the main one (Conjecture 6.3) pertaining to the covariance of our Lefschetz-Riemann-Roch map for proper morphisms of tame stacks (not necessarily representable). Other future works might be dedicated to the study of torsion in K'-groups as well as more general Artin stacks.