Tight closure, coherence, and localization at single elements

TL;DR

This paper introduces conditions for extending closure operations from rings to sheaves, examines their localization behavior, and applies these ideas to tight closure, revealing a new singularity type called semi F-regularity.

Contribution

It provides a geometric framework for understanding tight closure localization and introduces semi F-regularity, bridging existing singularity classes.

Findings

Conditions for extending closure operations to sheaves are established.

Tight closure satisfies these conditions, linking localization to quasi-coherence.

Semi F-regularity is introduced as a new intermediate singularity class.

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.