Pure subrings of Du Bois singularities are Du Bois singularities

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 Kovács 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 ${\underline{\Omega }}_X^0$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies.

19 pages. v5: Fixed typos, other small changes. v4: Rewrote Section 3, other small changes. v3: Extended to Noetherian $\mathbb{Q}$-algebras, added Theorem C. v2: Added part (ii) of Corollary B, other small changes