Valuation rings are derived splinters

Valuation rings of dimension one as limits of smooth algebras

“…This result is an application of [15,Theorem 1.2.5] which relies on some results from [3]. Also E. Elmanto and M. Hoyois proved that an absolute integrally closed valuation ring of residue field of characteristic p > 0 is a filtered direct limit of its regular finitely generated Z-subalgebras (see [1,Corollary 4.2.4]).…”

“…This local form of resolution of singularities remains open in positive and mixed characteristic, and implies that every valuation ring V should be a filtered direct limit of regular rings. There exists several nice extensions of Zariski's Uniformization Theorem as for example recently the result of B. Antieau, R. Datta [1,Theorem 4.1.1], which says that every perfect valuation ring of characteristic p > 0 is a filtered direct limit of its smooth F psubalgebras. This result is an application of [15,Theorem 1.2.5] which relies on some results from [3].…”

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).…”

Permanence properties of splinters via ultrapower

“…For Theorem 1.4, we dominate the ring R ′ with an ultrapower of R which we introduce and study in an ad-hoc fashion in Section 3. For the splinter property to be well-defined and preserved, we need to define and study non-Noetherian splinters in Section 2, which is partially done in [AD21]. Having obtained Theorem 1.4, Theorem 1.3 follows from taking completions and applying a standard commutative algebra argument.…”