The Grothendieck group of algebraic stacks

TL;DR

This paper develops a Grothendieck group for algebraic stacks, extends invariants like the Euler characteristic, and demonstrates that certain torsors have classes that defy simple product decomposition in the motivic setting.

Contribution

It introduces a Grothendieck group for algebraic stacks, identifies it with a localized version of the variety Grothendieck group, and extends invariants to distinguish classes in this new context.

Findings

Extended the Euler characteristic to algebraic stacks.

Defined invariants capable of distinguishing classes with same Euler characteristic.

Showed existence of $ ext{PSL}_n$-torsors with non-product classes in the motivic ring.

Abstract

We introduce a Grothendieck group of algebraic stacks (with affine stabilisers) analogous to the Grothendieck group of algebraic varieties. We then identify it with a certain localisation of the Grothendieck group of algebraic varieties. Several invariants of elements in this group are discussed. The most important is an extension of the Euler characteristic (of cohomology with compact support) but in characteristic zero we introduce invariants which are able to distinguish between classes with the same Euler characteristic. These invariants are actually defined on the completed localised Grothendieck ring of varieties used in motivic integration. In particular we show that there are $\PSL_n$-torsors of varieties whose class in the completed localised Grothendieck ring of varieties is not the product of the class of the base and the class of $\PSL_n$.