The Grothendieck group of algebraic stacks

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 $\PSL_n$.

21 pages; Two references have been added as well as a discussion of their relevance

Cited by in corpus (21)

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• Motivic classes of generalized Kummer schemes via relative power structures
• Homotopy Spectra and Diophantine Equations
• Stringy invariants and toric Artin stacks
• Deformed dimensional reduction
• Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve
• The Binomial Theorem and motivic classes of universal quasi-split tori
• On Chow-pure cohomology and Euler characteristics for motives and varieties, and their relation to unramified cohomology and Brauer groups
• The monodromy property for K3 surfaces allowing a triple-point-free model
• On the pre-lambda ring structure on the Grothendieck ring of stacks and the power structures over it
• A motivic change of variables formula for Artin stacks
• Arithmetic geometry of the moduli stack of Weierstrass fibrations over ${\mathbb{P}}^1$
• $\ell$-adic étale cohomology of the moduli of stable elliptic fibrations
• Motivic classes of classifying stacks of finite groups and unramified cohomology
• On the Mixed Tate property and the motivic class of the classifying stack of a finite group
• On the Motivic Class of the Moduli Stack of Twisted $G$-Covers