# zbMATH — the first resource for mathematics

Quotients by groupoids. (English) Zbl 0881.14018
Let $$G$$ be an algebraic group scheme of finite type over a locally noetherian base scheme $$L$$ and assume that $$G$$ acts morphically on the algebraic space $$X$$ of finite type over $$L$$. Following D. Mumford’s “Geometric invariant theory” (1965; Zbl 0147.39304) one defines the notions of geometric and (uniform) categorical quotients for the action of $$G$$ on $$X$$. As a consequence of their main result the authors obtain: If $$G$$ is flat over $$L$$ and acts properly on $$X$$ with finite stabilizer, then there is a geometric and uniform categorical quotient $$q : X \to Y$$ for the action of $$G$$ on $$X$$, where $$Y$$ is a separated algebraic space over $$Y$$. The presented approach requires the more general setting of groupoids; examples of such groupoids are obtained by restricting the equivalence relation defined by the action of $$G$$ on $$X$$ to a not necessarily $$G$$-stable subspace $$W$$ of $$X$$. In these terms the main result reads as follows:
Every flat groupoid $$\jmath : R \to X \times X$$ with finite stabilizer has a geometric and uniform categorical quotient. If $$\jmath$$ is finite, then the quotient space is separated.
As the authors outline, the concept of their proof is to choose appropriate subspaces $$W$$ of $$X$$ such that the restriction $$R'$$ of $$R$$ to $$W$$ is quasifinite and then reduce the problem to finding quotients for $$R' \to W \times W$$. The main result of the article under review generalizes various previous results on geometric quotients, see e.g. H. Popp [Invent. Math. 22, 1-40 (1973; Zbl 0281.14011)] and, more recently, E. Viehweg [“Quasi-projective moduli of polarized manifolds” (1995; Zbl 0844.14004)] and J. Kollár [Ann. Math., II. Ser. 145, No. 1, 33-79 (1997; see the preceding review)].

##### MSC:
 14M17 Homogeneous spaces and generalizations 14L30 Group actions on varieties or schemes (quotients) 14A20 Generalizations (algebraic spaces, stacks) 14L24 Geometric invariant theory
Full Text: