The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings

Abstract Suppose J = ( f 1 , … , f n ) $J = (f_1, \dots , f_n)$ upper J equals left parenthesis f 1 comma ellipsis comma f Subscript n Baseline right parenthesis is an n -generated ideal in any ring R . We prove a general Briançon-Skoda-type containment relating the integral closure J n + k − 1 ― $\overline {J^{n+k-1}}$ upper J Superscript n plus k minus 1 Baseline overbar with ordinary powers J k $J^k$ upper J Superscript k . We prove that our result implies the full Briançon-Skoda containment J n + k − 1 ― ⊆ J k $\overline {J^{n+k-1}} \subseteq J^k$ upper J Superscript n plus k minus 1 Baseline overbar subset of or equal to upper J Superscript 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 J n + k ― ⊆ J k $\overline {J^{n+k}} \subseteq J^k$ upper J Superscript n plus k Baseline overbar subset of or equal to upper J Superscript k for Du Bois singularities and even for a characteristic-free generalization. Our Briançon-Skoda-type theorem also implies well-known closure-based Briançon-Skoda results J n + k − 1