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S-equivalence for algebraic stacks

Xucheng Zhang

Abstract

We generalize the notion of S-equivalence, previously defined for semistable vector bundles, to points in arbitrary algebraic stacks and use it to describe the identification of points when passing to the moduli space. As applications, we recover the classical S-equivalence by considering the identification of semistable vector bundles in the moduli space, and discuss its relation to separatedness of the moduli space.

S-equivalence for algebraic stacks

Abstract

We generalize the notion of S-equivalence, previously defined for semistable vector bundles, to points in arbitrary algebraic stacks and use it to describe the identification of points when passing to the moduli space. As applications, we recover the classical S-equivalence by considering the identification of semistable vector bundles in the moduli space, and discuss its relation to separatedness of the moduli space.

Paper Structure

This paper contains 8 sections, 29 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. S-equivalence
  4. Jordan-Hölder filtration
  5. Identification of points
  6. Application to moduli of semistable bundles
  7. Application to other moduli of bundles
  8. Relation to separatedness

Key Result

Theorem 1.1

Let $\mathscr{X}$ be a locally reductive algebraic stack, locally of finite type over a field $k$. Suppose $\mathscr{X}$ admits an adequate moduli space $\phi: \mathscr{X} \to X$. Then for any geometric points $x_1,x_2 \in \mathscr{X}(\kappa)$, the following are equivalent: In particular, quasi-S-equivalence is an equivalence relation on points of $\mathscr{X}$.

Theorems & Definitions (67)

  • Theorem 1.1: Theorem \ref{['thm:equivalence']}
  • Remark 1.2
  • Corollary 1.3: Corollary \ref{['cor:main']}
  • Corollary 1.4: Corollary \ref{['cor:usual-S']}
  • Remark 1.5
  • Theorem 1.6: Theorem \ref{['thm:equivalence-non-opp']} and Theorem \ref{['thm:open-S-comp']}
  • Corollary 1.7: Corollary \ref{['cor:non=opp-in-Coh']} and Corollary \ref{['cor:nonsep=id']}
  • Definition 2.1: Filtration, DHL2014 or MR3758902
  • Example 2.2
  • Definition 2.3: Quasi-S-equivalence
  • ...and 57 more