Openness of splinter loci in prime characteristic

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 étale finite extension. We also show that for a noetherian N-graded ring over a field, the homogeneous maximal ideal detects the splinter property.Corollary 1.0.5. (See Corollary 4.3.9) Let R = ∞ n=0 R n be a noetherian graded ring such that R 0 = k is a field. Let m := n>0 R n be the homogeneous maximal ideal of R. Then R is a splinter if and only if R m is a splinter.

Algebraic geometry in mixed characteristic

“…A noetherian commutative ring R is called a splinter if it satisfies the conclusion of the direct summand conjecture, i.e., it splits off as a module from every finite extension. This class of singularities, formally introduced in [172], has recently received renewed attention (e.g., [173,10,70,7]). An external reason to care about this notion is a major conjecture in F -singularity theory ([126, page 85], [127, page 640]): splinters in characteristic p are expected to be the same as strongly F -regular rings (see [174, end of §3] for a discussion).…”