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 extend the key injectivity theorem of Kov\'acs and Schwede to arbitrary Noetherian schemes of equal characteristic zero. 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.