Grothendieck rings of Artin n-stacks

Abstract

We introduce a Grothendieck ring of higher Artin stacks generalizing the Grothendieck ring of algebraic varieties. We show that this ring is not trivial by noticing that it factors the invariant "number of rational points over a finite field". We also introduce the notion of "special Artin stacks", which by definition have affine homotopy groups π_{i}, and furthermore unipotent for i>1. Our principal theorem states that the natural inclusion morphism from the Grothendieck ring of varieties to the Grothendieck ring of special Artin stacks is an isomorphism after inverting the class of the affine line L and the classes of L^{i}-1 for all i>0. We deduce from this that several numerical invariants defined for varieties (e.g. Hodge numbers, l-adic Euler characteristic ...) extend uniquely to invariants defined for special Artin stacks. In particular we obtain a trace formula for special Artin stacks of finite type over a finite field, identifying the number of rational points as the trace of the Frobenius acting on the l-adic Euler characteristic.