The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings
Linquan Ma, Peter M. McDonald, Rebecca R. G., Karl Schwede
TL;DR
The paper proves a general Briançon-Skoda-type containment relating $\overline{J^{n+k-1}}$ to $J^k$ in all rings by working with a derived BE/Eagon-Northcott framework on partially normalized blowups. It unifies and strengthens known results for pseudo-rational, birational derived splinter, and Du Bois singularities, and extends these to characteristic-free settings, including perfectoid rings. The authors also connect the main containment to various closure operations, establishing characteristic-zero Hir closures and positive/mixed-characteristic plus/extended closures as natural consequences. As a major application, they establish uniform Briançon-Skoda and uniform Artin-Rees theorems for excellent (resp. excellent reduced) rings of finite dimension, resolving longstanding conjectures of Huneke. The work blends derived-category methods, explicit complexes on blowups, and modern uniformity techniques to produce broad, robust BS-type results with wide applicability.
Abstract
Suppose $J = (f_1, \dots, f_n)$ is an $n$-generated ideal in any ring $R$. We prove a general Briançon-Skoda-type containment relating the integral closure $\overline{J^{n+k-1}}$ with ordinary powers $J^k$. We prove that our result implies the full Briançon-Skoda containment $\overline{J^{n+k-1}} \subseteq J^k$ for pseudo-rational singularities (for instance regular rings), and even for the weaker condition of birational derived splinters. Our methods also yield the containment $\overline{J^{n+k}} \subseteq J^k$ for Du Bois singularities and even for a characteristic-free generalization. Our \myBrianconSkoda-type theorem also implies well-known closure-based Briançon-Skoda results $\overline{J^{n+k-1}} \subseteq (J^k)^{\mathrm{cl}}$ where, for instance, $\mathrm{cl}$ is tight or plus closure in characteristic $p > 0$, or $\mathrm{ep}$ closure or extension and contraction from $\widehat{R^+}$ in mixed characteristic. Our proof relies on a study of the tensor product of the derived image of the structure sheaf of a partially normalized blowup of $J$ with the Buchsbaum-Eisenbud complex (equivalently the Eagon-Northcott complex) associated to $(f_1,\dots,f_n)^k$. As an application of our results and methods above, we prove the uniform Artin-Rees theorem and the uniform Briançon-Skoda theorem for excellent, respectively excellent reduced, rings of finite dimension, answering conjectures of Huneke.