On some permanence properties of (derived) splinters

TL;DR

This paper investigates the stability of the splinter property in Noetherian rings under various algebraic operations, providing new results and examples that support a conjecture about regular maps preserving splinters.

Contribution

It proves that Noetherian splinters ascend under essentially étale homomorphisms and that certain localizations and completions of splinters remain splinters, advancing understanding of their permanence properties.

Findings

Noetherian splinters ascend under essentially étale homomorphisms.

Henselization of a local splinter is a splinter.

Completion of a local splinter with geometrically regular fibers is a splinter.

Abstract

We show that Noetherian splinters ascend under essentially \'etale homomorphisms. Along the way, we also prove that the henselization of a Noetherian local splinter is always a splinter and that the completion of a local splinter with geometrically regular formal fibers is a splinter. Finally, we give an example of a (non-excellent) Gorenstein local splinter with mild singularities whose completion is not a splinter. Our results provide evidence for a strengthening of the direct summand theorem, namely that regular maps preserve the splinter property.