Equidimensional morphisms onto splinters are pure

TL;DR

This paper characterizes splinter rings via equidimensional morphisms and establishes purity results for certain classes of ring maps and schemes, with implications for $F$-rationality.

Contribution

It introduces a new characterization of splinter rings using equidimensional morphisms and proves purity results for fibrations over specific schemes.

Findings

A Noetherian ring is a splinter iff all equidimensional surjective morphisms are pure.

Equidimensional fibrations over normal $ extbf{Q}$-schemes or regular schemes are strongly pure.

A weak Boutot-type theorem shows $F$-rationality descends under certain pure, equidimensional maps.

Abstract

We prove that a Noetherian ring $R$ is a splinter if and only if for every equidimensional surjective morphism $\operatorname{Spec}(S) \to \operatorname{Spec}(R)$, the map $R \to S$ is pure. This yields a large, nontrivial class of ring maps that are automatically pure. More generally, we prove that a locally Noetherian scheme $Y$ is locally a splinter if and only if every locally equidimensional morphism $X \to Y$ is strongly pure. Special cases of our results show that equidimensional fibrations over normal $\mathbf{Q}$-schemes or regular schemes of arbitrary characteristic are strongly pure. The main ingredient is a new factorization result for locally equidimensional morphisms of schemes, which is of independent interest. Additionally, we prove a weak Boutot-type theorem for $F$-rationality, which says that $F$-rationality descends under pure ring maps that are locally equidimensional under universally catenary assumptions. This statement is false without the locally equidimensional hypothesis.