Quotients by groupoids.

*(English)*Zbl 0881.14018Let \(G\) 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)].

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)].

Reviewer: J.Hausen (Konstanz)