Equidimensional morphisms onto splinters are pure

Abstract

We prove that a Noetherian ring $R$ is a splinter if and only if for every equidimensional surjective morphism $\mathrm{Spec}(S)\to \mathrm{Spec}(R)$, 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.

13 pages. v2: Added references, fixed typos