Grothendieck rings of Artin n-stacks

TL;DR

This paper introduces a Grothendieck ring for higher Artin stacks, generalizing the variety case, and proves an isomorphism with the variety Grothendieck ring after certain localizations, enabling extension of numerical invariants.

Contribution

It defines the Grothendieck ring of higher Artin stacks and proves an isomorphism with the variety Grothendieck ring after inverting specific classes, extending invariants to stacks.

Findings

The Grothendieck ring of higher Artin stacks is non-trivial.

The natural map from the variety Grothendieck ring to that of special Artin stacks becomes an isomorphism after localization.

Numerical invariants like Hodge numbers extend to special Artin stacks, with a trace formula for rational points.

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 \pi_{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.