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© The London Mathematical Society
Constructible Functions on Artin Stacks
The Mathematical Institute 24–29 St. Giles', Oxford OX1 3LB, United Kingdom joyce{at}maths.ox.ac.uk
Received 13 April 2005. Revision received 10 January 2006.
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() 
has f(X()) finite and f–1(c) constructible for all c 
0. Write CF(X) for the vector space of such f. Let 
: X Y and 
: Y Z be morphisms of 
-varieties. MacPherson defined a linear pushforward CF() : CF(X) CF(Y) by ‘integration’ with respect to the topological Euler characteristic. It is functorial, that is, CF( 
) = CF() CF(). 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.