The Grothendieck group of algebraic stacks

Details The Grothendieck group of algebraic stacks The Grothendieck group of algebraic stacks

2009-03-18

arxiv.org/abs/0903.3143v2

Mathematics

Algebraic Geometry

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

Scientific paper

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 $\PSL_n$.

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