Pure subrings of Du Bois singularities are Du Bois singularities
Charles Godfrey, Takumi Murayama
TL;DR
The paper establishes a Boutot-type descent: if $R \to S$ is cyclically pure and $S$ has Du Bois singularities in the $h$ topology, then $R$ inherits Du Bois singularities as well. It develops a characteristic-free framework for Du Bois theory via Grothendieck topologies, proving a key injectivity theorem in equal characteristic zero using Zariski–Riemann spaces and a Hodge-theoretic input, which in turn yields descent results and lc-type corollaries when $K_R$ is Cartier. The approach unifies and extends existing results in characteristic zero and provides new insights in prime and mixed characteristic, linking Du Bois properties to injectivity criteria and to log canonical-type singularities. The work also broadens Kovács–Schwede-type injectivity methods to pairs and to a topological/limit-compactification setting, offering tools potentially applicable to broader descent problems in singularity theory and algebraic geometry.
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 $\underlineΩ^0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies.