Centers of F-purity

TL;DR

This paper introduces and studies centers of F-purity, positive characteristic analogues of centers of log canonicity, establishing new subadjunction results and characterizations of test ideals, with some results extending to characteristic zero.

Contribution

It develops the theory of centers of F-purity, proves new subadjunction-like results, and characterizes test ideals using uniformly F-compatible ideals, unifying previous concepts.

Findings

Established positive characteristic analogues of centers of log canonicity.

Proved new subadjunction-like results in F-purity context.

Characterized test ideals via uniformly F-compatible ideals.

Abstract

In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair $(R, \Delta)$. We call these analogues centers of $F$-purity. We prove positive characteristic analogues of subadjunction-like results, prove new stronger subadjunction-like results, and in some cases, lift these new results to characteristic zero. Using a generalization of centers of $F$-purity which we call uniformly $F$-compatible ideals, we give a characterization of the test ideal (which unifies several previous characterizations). Finally, in the case that $\Delta = 0$, we show that uniformly $F$-compatible ideals coincide with the annihilators of the $\mathcal{F}(E_R(k))$-submodules of $E_R(k)$ as defined by Smith and Lyubeznik.