Openness of splinter loci in prime characteristic

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