The Grothendieck group of algebraic stacks
By Torsten Ekedahl. 2009
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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 $\PSL_n$.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/869005b6c0023755168ba2cbad8989570bd90b75
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In Algebra and Number Theory
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Citations in the corpus
- On the motives of moduli of chains and Higgs bundles (2011, background): “tion [X] = [X\Z]+[Z] whenever Z⊂ X is a closed subvariety. (See [30, Section 2] for background on this ring.) 1Other authors have considered similar definitions, see Joyce [24], To¨en [37] and Ekedahl [11]. 4 O. GARCIA-PRADA, J. HEINLOTH, AND A. SCHMITT One denotes the class of the affine line by L:= [”