CONSTRUCTIBLE FUNCTIONS ON ARTIN STACKS

Abstract

Let $\mathbb K$ be an algebraically closed field, let $X$ be a $\mathbb K$-variety, and let $X(\mathbb K)$ be the set of closed points in $X$. A constructible set$C$ in $X(\mathbb K)$ is a finite union of subsets $Y(\mathbb K)$ for subvarieties $Y$ in $X$. A constructible function$f:X(\mathbb K)\rightarrow\mathbb Q$ has $f(X(\mathbb K))$ finite and $f^{-1}(c)$ constructible for all $c\ne 0$. Write CF$(X)$ for the vector space of such $f$. Let $\phi:X\rightarrow Y$ and $\psi: Y\rightarrow Z$ be morphisms of ${\mathbb C}$-varieties. MacPherson defined a linear pushforward CF$(\phi):{\rm CF}(X)\rightarrow{\rm CF}(Y)$ by ‘integration’ with respect to the topological Euler characteristic. It is functorial, that is, CF$(\psi\circ\phi)={\rm CF}(\psi)\circ{\rm CF}(\phi)$. This was extended to $\mathbb K$ of characteristic zero by Kennedy.

This paper generalizes these results to $\mathbb K$-schemes and Artin$\mathbb K$-stacks with affine stabilizer groups. We define the notions of Euler characteristic for constructible sets in $\mathbb K$-schemes and $\mathbb K$-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.