Constructible functions on Artin stacks - Joyce, D (2004)
Let K be an algebraically closed field, X a K-scheme, and X(K) the set of closed points in X. A constructible set C in X(K) is a finite union of subsets Y(K) for finite type subschemes Y in X. A constructible function f : X(K) --> Q has f(X(K)) finite and f^{-1}(c) constructible for all nonzero c. Write CF(X) for the Q-vector space of constructible functions on X. Let phi : X --> Y and psi : Y --> Z be morphisms of C-varieties. MacPherson defined a Q-linear "pushforward" CF(phi) : CF(X) --> CF(Y) by "integration" w.r.t. the topological Euler characteristic. It is functorial, that is, CF(psi o phi)=CF(psi) o CF(phi). This was extended to K of characteristic zero by Kennedy. This paper generalizes these results to K-schemes and Artin K-stacks with affine stabilizers. We define notions of Euler characteristic for constructible sets in K-schemes and K-stacks, and pushforwards and pullbacks of constructible functions, with functorial behaviour. Pushforwards and pullbacks commute in Cartesian squares. We also define "pseudomorphisms", a generalization of morphisms well suited to constructible functions problems.