Openness of splinter loci in prime characteristic

TL;DR

This paper investigates the openness of splinter loci in prime characteristic schemes, establishing conditions under which the splinter property is open and connecting it to $F$-compatibility and Frobenius splitting.

Contribution

It proves that under certain conditions, the splinter locus in prime characteristic schemes is open, and links splinter properties to Frobenius splitting and $F$-compatible ideals.

Findings

Splinter locus is open in schemes with finite Frobenius or over quasi-excellent rings.

Splinter property is detected by a single generically étale extension in rings with pure Frobenius.

Homogeneous maximal ideal detects splinter property in graded rings over fields.

Abstract

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and $F$-compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically \'etale finite extension. We also show that for a noetherian $\textbf{N}$-graded ring over a field, the homogeneous maximal ideal detects the splinter property.