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Intersection theory on algebraic stacks and on their moduli spaces. (English) Zbl 0694.14001
This article develops the theory of algebraic stacks due to Deligne and Artin, extending the notion of scheme, from ground level up to results:
(1) Showing an algebraic stack having a moduli space admits a finite ramified covering by a scheme;
(2) Characterizing those schemes of finite type over a field of characteristic 0 which are moduli spaces of some smooth stack;
(3) Proving that Gysin homomorphisms for regular local embeddings of stacks pass to rational equivalence and commute among themselves;
(4) Showing that the Chow group of a moduli space of an algebraic stack is isomorphic to the Chow group of the stack (despite the introduction).
The moduli space of a smooth algebraic stack of finite type over a field is then shown to be an Alexander scheme (satisfying most of the formal intersection properties that smooth schemes do).
Key inputs are intersection theory à la Fulton-MacPherson; Gillet’s notion of a moduli space for an algebraic stack and Gillet’s definition of Chow group for an algebraic stack of finite type of a field.
The article is roughly speaking derived from the author’s doctoral thesis.
Reviewer: P.Cherenack

MSC:
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
14F99 (Co)homology theory in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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References:
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