Effective morphisms and quotient stacks

Abstract

We give a valuative criterion for when a smooth algebraic stack with a separated good moduli space is the quotient of a separated Deligne-Mumford stack by a torus. For doing so, we introduce a new class of morphisms, the so-called effective morphisms, which are a generalization of separated morphisms.

arXiv comment25 pages, comments are welcome! v2: we corrected a gap in a proof and slightly modified the definition of effective morphism. To appear on IMRN