Equidimensional morphisms onto splinters are pure
Takumi Murayama
TL;DR
The paper establishes that splinterness of a Noetherian ring is equivalent to purity of all equidimensional surjections onto its spectrum, and extends this to a scheme-level criterion where locally equidimensional morphisms are strongly pure exactly when the base is locally a splinter. A new factorization theorem for locally equidimensional morphisms underpins these results, enabling purity arguments without Bertini-type tools. The work also proves a weak Boutot-type descent for F-rationality under locally equidimensional, universally catenary conditions, and discusses implications for equidimensional fibrations over normal Q-schemes or regular bases. Overall, it broadens the class of morphisms for which purity and descent properties hold, and provides a robust framework for future descent questions in singularity theory.
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.