Pure subrings of Du Bois singularities are Du Bois singularities

TL;DR

This paper proves that subrings of Noetherian $Q$-algebras with Du Bois singularities also have Du Bois singularities, extending known results and providing new insights in various characteristic settings.

Contribution

It establishes that pure subrings inherit Du Bois singularities, even in cases of cyclically pure maps, and connects this to log canonical singularities under certain conditions.

Findings

Subrings of Du Bois singularities are Du Bois.

Results extend to prime and mixed characteristic.

Connections to log canonical singularities are established.

Abstract

Let $R \to S$ be a cyclically pure map of Noetherian $\mathbb{Q}$-algebras. In this paper, we show that if $S$ has Du Bois singularities, then $R$ has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. Our proof also yields interesting results in prime characteristic and in mixed characteristic. As a consequence, we show that if $R \to S$ is a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb{C}$, $S$ has log canonical type singularities, and $K_R$ is Cartier, then $R$ has log canonical singularities. Along the way, we prove a version of the key injectivity theorem of Kov\'acs and Schwede for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. Throughout the paper, we use the characterization of the complex $\underline{\Omega}^0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies.