Openness of splinter loci for arbitrary excellent schemes: the mixed-characteristic and general prime-characteristic cases

A noetherian ring is a splinter if it is a direct summand of every finite extension ring. Datta and Tucker proved that the splinter locus of a locally noetherian scheme of prime characteristic is open in the two large classes they treat (F-finite schemes, and schemes locally essentially of finite type over a noetherian local G-ring of characteristic p), and in equal characteristic zero openness is immediate because splinters coincide with normal rings. They state the expectation that splinter loci should be open for arbitrary excellent schemes in any characteristic. The expectation remains unproved: no work settles openness for excellent mixed-characteristic schemes (for instance spectra of rings essentially of finite type over the p-adic integers) or for excellent prime-characteristic schemes outside the two treated classes, and the parallel openness question for the adjacent property of birational derived splinters is likewise explicitly open (Lyu, Question 8.1).

Background

A noetherian ring $R$ is called a splinter if every finite ring map $R\to S$ that is surjective on spectra admits an $R$-linear left inverse; equivalently, $R$ is a direct summand, as an $R$-module, of every module-finite extension ring. The assertion that regular noetherian rings are splinters is the direct summand conjecture, now a theorem in all characteristics (settled in mixed characteristic around 2016-2018 by perfectoid methods), and it makes the splinter condition a fundamental notion of singularity. Splinters are always normal, and conversely every normal noetherian $\mathbf{Q}$-algebra is a splinter, so the notion is substantive precisely in prime and mixed characteristic. For a locally noetherian scheme $X$ one accordingly studies the splinter locus

$$\operatorname{Spl}(X)=\{x\in X:\ \mathcal{O}_{X,x}\ \text{is a splinter}\}.$$

For the local ring properties usually met in geometry and arithmetic (normal, Cohen-Macaulay, Gorenstein, complete intersection) the corresponding loci are open on the schemes one encounters, and asking the same for the splinter locus is a basic question about these singularities; Datta and Tucker single it out as one of the outstanding unresolved issues. In equal characteristic zero the question is immediate: there the splinter locus is the normal locus, and the normal locus of a quasi-excellent scheme is open. (Excellence, and its weakening quasi-excellence, are standard regularity conditions on formal fibers and regular loci, satisfied by schemes of finite type over fields, over the integers, and over complete local rings; virtually all schemes arising in arithmetic geometry are excellent.) In prime and mixed characteristic the question is genuinely open beyond the classes settled by Datta and Tucker.

Their theorem covers schemes of prime characteristic $p>0$ that are either $F$-finite (the Frobenius endomorphism $x\mapsto x^p$ makes the ring a finite module over its image, as for varieties over perfect fields) or locally essentially of finite type over a noetherian local ring of characteristic $p$ whose formal fibers are geometrically regular (the G-ring condition underlying excellence); this includes all varieties over fields of positive characteristic and complete local rings of characteristic $p$. Their method shapes what remains open: for an $F$-pure domain $R$ (Frobenius is pure, i.e. universally injective), the trace ideals $\tau_{S/R}=\operatorname{im}(\operatorname{Hom}_R(S,R)\to R)$ of the finite subalgebras $S$ of the absolute integral closure $R^{+}$ (the integral closure of $R$ in an algebraic closure of its fraction field) are uniformly $F$-compatible ideals, that is, ideals stable under all Frobenius-compatible maps, a prime-characteristic analogue of centers of log canonicity. In their classes there are only finitely many such ideals, the traces stabilize, and the non-splinter locus is the closed set $V(\tau_R)$ cut out by a single trace ideal. The authors write: "We expect the splinter loci to be open for arbitrary excellent schemes in any characteristic," while noting that excellence-type hypotheses cannot be dropped entirely: by a meta construction of Hochster there exist locally excellent (but not excellent) noetherian domains whose splinter loci are not open.

The difficulty is that this whole mechanism is Frobenius-based. In prime and mixed characteristic, excellent local splinters are known to be Cohen-Macaulay and pseudo-rational (a characteristic-free strengthening of rational singularities), and in prime characteristic splinters are conjectured to coincide with strongly $F$-regular rings, the tight-closure-theoretic analogue of Kawamata log terminal singularities (tight closure being an ideal-closure operation defined via the Frobenius); this conjecture is known for Gorenstein rings, for $\mathbf{Q}$-Gorenstein rings (Singh), and for rings with noetherian anti-canonical cover. In mixed characteristic, where none of the Frobenius machinery exists, splinter-type conditions are currently approached through perfectoid techniques: mixed-characteristic splinters are Cohen-Macaulay and have rational singularities, and an active line of work develops perfectoid pure and perfectoid injective singularities (purity of the ring into a suitable perfectoid, i.e. p-adically complete and 'almost perfect', extension, as analogues of $F$-pure and $F$-injective singularities). None of this delivers an analogue of the finite stabilization of traces, which is what openness of splinter loci in mixed characteristic would seem to require.

Problem Statement

Determine whether the splinter locus of every excellent noetherian scheme is open. Following the expectation stated by Datta and Tucker at the end of the introduction of "Openness of splinter loci in prime characteristic": is

$$\operatorname{Spl}(X)=\{x\in X:\ \mathcal{O}_{X,x}\ \text{is a splinter}\}$$

an open subset of $X$ for every excellent noetherian scheme $X$, in arbitrary characteristic?

The following cases are known: (a) equal characteristic zero, where quasi-excellence suffices because the splinter locus is the normal locus; (b) prime characteristic $p>0$, where openness holds when $X$ is $F$-finite or locally essentially of finite type over a noetherian local G-ring of characteristic $p$ (Datta and Tucker). The remaining open cases are mixed characteristic (the central one, for example $X=\operatorname{Spec}R$ with $R$ essentially of finite type over $\mathbf{Z}_p$ or over an excellent mixed-characteristic local ring such as a discrete valuation ring or a formal power series ring over one) and excellent schemes of prime characteristic that are neither $F$-finite nor locally essentially of finite type over a noetherian local G-ring of characteristic $p$. An affirmative answer is a proof of openness for all excellent schemes in every characteristic; a negative answer is an excellent scheme together with a point of $\operatorname{Spl}(X)$ at which openness fails (non-splinter points in every neighborhood). Some excellence-type hypothesis is necessary: locally excellent non-excellent domains with non-open splinter locus exist by Hochster's meta construction.

References

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