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Tight closure, coherence, and localization at single elements

Neil Epstein

Abstract

In this note, a condition (\emph{open persistence}) is presented under which a (pre)closure operation on submodules (resp. ideals) over rings of global sections over a scheme $X$ can be extended to a (pre)closure operation on sheaves of submodules of a coherent $\mathcal{O}_X$-module (resp. sheaves of ideals in $\mathcal{O}_X$). A second condition (\emph{glueability}) is given for such an operation to behave nicely. It is shown that for an operation that satisfies both conditions, the question of whether the operation commutes with localization at single elements is equivalent to the question of whether the new operation preserves quasi-coherence. It is shown that both conditions hold for tight closure and some of its important variants, thus yielding a geometric reframing of the open question of whether tight closure localizes at single elements. A new singularity type (\emph{semi F-regularity}) arises, which sits between F-regularity and weak F-regularity. The paper ends with (1) a case where semi F-regularity and weak F-regularity coincide, and (2) a case where they cannot coincide without implying a solution to a major conjecture.

Tight closure, coherence, and localization at single elements

Abstract

In this note, a condition (\emph{open persistence}) is presented under which a (pre)closure operation on submodules (resp. ideals) over rings of global sections over a scheme can be extended to a (pre)closure operation on sheaves of submodules of a coherent -module (resp. sheaves of ideals in ). A second condition (\emph{glueability}) is given for such an operation to behave nicely. It is shown that for an operation that satisfies both conditions, the question of whether the operation commutes with localization at single elements is equivalent to the question of whether the new operation preserves quasi-coherence. It is shown that both conditions hold for tight closure and some of its important variants, thus yielding a geometric reframing of the open question of whether tight closure localizes at single elements. A new singularity type (\emph{semi F-regularity}) arises, which sits between F-regularity and weak F-regularity. The paper ends with (1) a case where semi F-regularity and weak F-regularity coincide, and (2) a case where they cannot coincide without implying a solution to a major conjecture.
Paper Structure (4 sections, 23 theorems, 3 equations)

This paper contains 4 sections, 23 theorems, 3 equations.

Table of Contents

  1. Persistence and closure operations on subsheaves
  2. Glueable (pre)closure operations
  3. Quasi-coherence and localization at elements
  4. Application to tight closure and its variants

Key Result

Lemma 1.6

Let ${\mathrm{cl}}$ be an openly persistent preclosure operation on modules over a scheme $X$. Let $\mathcal{M}$ be a quasi-coherent $\mathcal{O}_X$-module, and $\mathcal{L}$ a sheaf of $\mathcal{O}_X$-submodules of $\mathcal{M}$. Let $U$, $V$ be affine open subsets of $X$ with $V \subseteq U$. Then

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Example 1.5
  • Lemma 1.6
  • proof
  • proof
  • Remark 1.8
  • Definition 2.1
  • ...and 47 more