Constructible Functions on Artin Stacks

Let be an algebraically closed field, let X be a ‐variety, and let X () be the set of closed points in X . A constructible set C in X () is a finite union of subsets Y () for subvarieties Y in X . A constructible function f : X () → ℚ has f ( X ()) finite and f −1 ( c ) constructible for all c ≠ 0. Write CF( X ) for the vector space of such f . Let φ : X → Y and ψ : Y → Z be morphisms of ℂ‐varieties. MacPherson defined a linear pushforward CF(φ) : CF( X ) → CF( Y ) by ‘integration’ with respect to the topological Euler characteristic. It is functorial, that is, CF(ψ ○ φ) = CF(ψ) ○ CF(φ). This was extended to of characteristic zero by Kennedy. This paper generalizes these results to ‐ schemes and Artin ‐ stacks with affine stabilizer groups. We define the notions of Euler characteristic for constructible sets in ‐schemes and ‐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.