The Quarterly Journal of Mathematics Advance Access originally published online on June 2, 2007
The Quarterly Journal of Mathematics 2007 58(3):345-392; doi:10.1093/qmath/ham019
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MOTIVIC INVARIANTS OF ARTIN STACKS AND ‘STACK FUNCTIONS’
The Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, UK
E-mail: joyce{at}maths.ox.ac.uk
Received 31 January 2006;
revised 15 December 2006
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Abstract |
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An invariant 
of quasiprojective 
-varieties X with values in a commutative ring 
is motivic if (X) = (Y) + (X\ Y) for Y closed in X, and (X x Y) = (X)(Y). Examples include Euler characteristics 
and virtual Poincaré and Hodge polynomials. We first define a unique extension ' of to finite type Artin -stacks 
, which is motivic and satisfies '([X/G]) = (X)/(G) when X is a -variety, G a special -group acting on X, and [X/G] is the quotient stack. This only works if (G) is invertible in for all special -groups G, which excludes = as (
m) = 0. But we can extend the construction to get round this.
Then we develop the theory of stack functions on Artin stacks. These are a universal generalization of constructible functions on Artin stacks. There are several versions of the construction: the basic one 
, and variants 
‘twisted’ by motivic invariants. We associate a 
-vector space or a -module 
to each Artin stack , with functorial operations of multiplication, pullbacks 
* and pushforwards * under 1-morphisms 
;, and so on. They will be important tools in the author's series on ‘Configurations in abelian categories’.