Remarks on effective uniform Briançon-Skoda
Alexandria Wheeler, Wenliang Zhang
TL;DR
The paper addresses explicit uniform Briançon–Skoda bounds for Noetherian local rings by linking the uniform exponent $k$ to invariants such as analytic spread $\ell(I)$ and height $\operatorname{ht}(I)$ under $F$-pure and related conditions, as well as in characteristic $0$ via dense $F$-pure type. The main results show that in prime characteristic $p$, $\overline{I^{\ell+n}}\subseteq (I^n)^F$ for $I$ generated by $\ell$ elements, and, when $R$ is $F$-pure, $\overline{I^{\ell+n}}\subseteq I^n$; with infinite residue field, one can take $I$ generated by $\ell(I)$ elements to obtain $\overline{I^{\ell(I)+n}}\subseteq I^n$, with additional bounds such as $\overline{I^{2\ell(I)-\operatorname{ht}(I)+1}}\subseteq I$ under further hypotheses and $\overline{I^{\dim(R)+n}}\subseteq I^n$ for Cohen–Macaulay $F$-injective rings. In characteristic $0$, analogous uniform bounds hold for rings of dense $F$-pure type, extending the reach of Skoda-type results beyond pseudo-rational cases and tying the bounds to reductions and singularity types like log canonical and Du Bois. Overall, the work provides explicit, invariant-driven exponents that guarantee containment of integral closures in powers of $I$, with sharpness considerations and open questions for broader classes.
Abstract
Let $R$ be a noetherian commutative ring. Of great interest is the question whether one can find an explicit integer $k$ such that $\overline{I^{k+n}}\subseteq I^n$ for each ideal $I$ and each integer $n\geq 1$ (the notation $\overline{I^{k+n}}$ denotes the integral closure of $I^{k+n}$). In this article, we investigate this question and obtain optimal values of $k$ for $F$-pure (or dense $F$-pure type) rings and Cohen-Macaulay $F$-injective (or dense $F$-injective type) rings.