Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Effective morphisms and quotient stacks

Andrea Di Lorenzo, Giovanni Inchiostro

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.

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.
Paper Structure (14 sections, 35 theorems, 37 equations)

This paper contains 14 sections, 35 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Effective morphisms
  4. The "local bug-eyed cover”
  5. Effective morphisms
  6. Preparatory lemmas on Picard groups
  7. Effectiveness of quotients of DM stacks by a torus
  8. Torsors, twisted tori and gerbes
  9. Torsors
  10. Twisted tori and gerbes
  11. When $\mathcal{B} G$ is not effective
  12. From effective to the quotient presentation
  13. Base step: effective good moduli spaces of dimension zero
  14. Inductive step: extending a generating set

Key Result

Theorem 1.1

Let $\mathcal{X}$ be a smooth algebraic stack and $\pi:\mathcal{X}\to X$ a good moduli space morphism. Assume that $X$ is separated, over a field $k$ of characteristic 0 containing all the roots of 1. Then $\pi$ is effective if and only $\mathcal{X}$ is the quotient of a separated Deligne-Mumford st

Theorems & Definitions (82)

  • Theorem 1.1: \ref{['teo quotient implies effective']} and \ref{['teo effettivo implica ho il gen set']}
  • Theorem 1.2: \ref{['teo val criterion']}
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Proposition 2.6
  • Lemma 2.8
  • proof
  • ...and 72 more