Openness of splinter loci in prime characteristic (Q6155815)

In this nice opus, the authors consider splinters and the splinter locus of an affine scheme of characteristic \(p>0\). A ring is splinter if it is a direct summand of every finite cover. When \(R\) is either \(F\)-finite or local, the authors show that there are finitely many uniformly \(F\)-compatible ideals, among them the trace, \(\tau_R\), and the ideal trace, \(T_R\). The authors compare these ideals to test ideals of module closures \(\text{cl}_B\) and the tight closure test ideals, both big and finitistic. When \(R\) is a Noetherian domain, the authors show that \(R\) is splinter if and only if \(\tau_R=R=T_R\). They further show that if \(R\) is an \(F\)-pure noetherian normal domain which is either \(F\)-finite, local or of essentially finite type over a noetherian local ring with geometrically regular formal fibers, then the splinter locus is open and is the complement of \(V(\tau_R)\).