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

doi:10.1017/fmp.2026.10035

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

Part of: Birational geometry Commutative algebra: Homological methods Local theory General commutative ring theory Ring extensions and related topics

Published online by Cambridge University Press: 15 July 2026

and

Linquan Ma: Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA; E-mail: ma326@purdue.edu
Peter M. McDonald: Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada; E-mail: pma94@sfu.ca
Rebecca R.G.: Affiliation:
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA; E-mail: rrebhuhn@gmu.edu
Karl Schwede*: Affiliation:
Department of Mathematics, University of Utah, Salt Lake City UT, 84112, USA
*: E-mail: schwede@math.utah.edu (Corresponding author)

Article contents

Abstract
Introduction
Integral closure and the Buchsbaum-Eisenbud complex
Generalizations of rational and Du Bois singularities
Connections with closure operations
Uniform Briançon-Skoda and uniform
Competing interests
Funding statement
Footnotes
References

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 Briançon-Skoda-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 quasi-excellent, respectively quasi-excellent reduced, rings of finite dimension, answering conjectures of Huneke.

MSC classification

Primary: 13B22: Integral closure of rings and ideals ; integrally closed rings, related rings (Japanese, etc.)

Secondary: 14B05: Singularities 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 13D02: Syzygies, resolutions, complexes 14E15: Global theory and resolution of singularities

Information

Type: Algebra
Information: Forum of Mathematics, Pi , Volume 14 , 2026 , e19

DOI: https://doi.org/10.1017/fmp.2026.10035 [Opens in a new window]
Creative Commons: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright: © The Author(s), 2026. Published by Cambridge University Press

1 Introduction

Suppose R is a ring and $J = (f_1, \dots , f_n)$ is an ideal. In [Reference Skoda and BriançonSB74] analytic methods were used to show the famous Briançon-Skoda theorem, namely that

$$\begin{align*}\overline{J^{n+k-1}} \subseteq J^k \end{align*}$$

for all $k\in \mathbb {N}$ if R is the coordinate ring of a smooth variety over $\mathbb {C}$ and where $\overline {(-)}$ denotes integral closure. This was generalized to all regular rings in [Reference Lipman and SathayeLS81]. Alternate proofs and improvements in various cases can be found for instance in [Reference Aberbach and HunekeAH93, Reference LipmanLip94, Reference Aberbach, Huneke and TrungAHT95, Reference HyryHVU98, Reference Ein and LazarsfeldEL99, Reference SchoutensSch03a, Reference AnderssonAnd06, Reference SznajdmanSzn10]. There have been many such results for singular rings as well. For many Noetherian ringsFootnote ¹ , Huneke proved that there exists some integer m such that $\overline {J^{m+k}} \subseteq J^k$ for all ideals $J\subseteq R$ ; this is the so-called uniform Briançon-Skoda theorem [Reference HunekeHun92] (cf. [Reference HunekeHun00, Reference Andersson, Samuelsson and SznajdmanASS10, Reference SznajdmanSzn18, Reference Cid-Ruiz and JeffriesCRJ25, Reference Katz and PolstraKP25]). He then used this to prove a uniform Artin-Rees theorem for these rings, namely for each containment of finitely generated R-modules $N \subseteq M$ , there exists an integer $\ell $ such that for all ideals $I\subseteq R$ , we have $I^kM \cap N \subseteq I^{k-\ell } M$ for all $k\geq \ell $ (cf. [Reference Eisenbud and HochsterEH79, Reference O’CarrollO’C87, Reference Duncan and O’CarrollDO89, Reference WangWan97]). We will prove such a uniform Artin-Rees and uniform Briançon-Skoda theorem for all excellent (resp. excellent reduced) rings of finite dimension, resolving Huneke’s conjectures [Reference HunekeHun92, Conjecture 1.3 and Conjecture 1.4].

On the more precise side, Lipman-Teissier [Reference Lipman and TeissierLT81] proved that $\overline {J^{\dim R+k-1}} \subseteq J^k$ if the ambient ring R has pseudo-rational singularities (a characteristic-free version of rational singularities); compared with the nonsingular case, one notes that n was replacedFootnote ² by $\dim R$ , so this statement is not quite optimal. Inspired by [Reference Lipman and TeissierLT81], Aberbach-Huneke [Reference Aberbach and HunekeAH01] proved the full version of the Briançon-Skoda theorem, namely that $\overline {J^{n+k-1}} \subseteq J^k$ , for excellent F-rational rings (a characteristic $p> 0$ analog of rational singularities which are pseudo-rational by [Reference SmithSmi97]). This implies that $\overline {J^{n+k-1}} \subseteq J^k$ for rational singularities in characteristic zero via reduction mod $p \gg 0$ [Reference HaraHar98, Reference Mehta and SrinivasMS97], see also [Reference LiLi21]. However, it was unclear whether the full bound holds for pseudo-rational rings in positive or mixed characteristic. Below, in Corollary A, we obtain the full version of the Briançon-Skoda theorem for generalizations of pseudo-rational singularities.

For sufficiently singular rings, it is not always the case that $\overline {J^{n+k-1}} \subseteq J^k$ . However, Hochster-Huneke showed in [Reference Hochster and HunekeHH90] that

$$\begin{align*}\overline{J^{n+k-1}} \subseteq (J^k)^* \end{align*}$$

assuming R is a Noetherian ring of characteristic $p> 0$ and where $(-)^*$ denotes the tight closure, a way to slightly enlarge the ideal – a trivial operation if the ring has sufficiently mild singularities. Related results can then be deduced in equal characteristic zero via reduction mod $p \gg 0$ , see [Reference Hochster and HunekeHH06, Reference SchoutensSch03b, Reference Aschenbrenner and SchoutensAS07, Reference BrennerBre03]. Back in characteristic $p> 0$ , the tight closure Briançon-Skoda theorem was improved to the plus closure version in [Reference Hochster and HunekeHH95]. Analogous closure related mixed characteristic results can be found in [Reference HeitmannHei97, Reference HeitmannHei01, Reference Heitmann and MaHM21, Reference Ma, Schwede, Tucker, Waldron and WitaszekMST+22, Reference Rodríguez-Villalobos and SchwedeRS24] the last of which utilizes [Reference BhattBha20].

Our first main theorem, which has a remarkably simple proof, implies, strengthens and unifies many of the results above.

Main Theorem (Theorem 2.2)

Suppose R is a ring and $J = (f_1, \dots , f_n)$ is an ideal. Set $Y \to \mathrm {Spec} \, R$ to be the blowup of $\overline {J^{n+k-1}}$ (or any map that dominates it, such as the normalized blowup of J in $\mathrm {Spec} \, R$ if R is reduced and has finitely many minimal primesFootnote ³ ). Then $\overline {J^{n+k-1}}$ maps to zero in

$$\begin{align*}H_0\big( L^k(\underline{f}) {\otimes}^{{\mathbb L}} {\mathbb R} \Gamma(Y, \mathcal{O}_{Y})\big) \end{align*}$$

where $L^k(\underline {f})$ is the Buchsbaum-Eisenbud L-complex associated to $(f_1, \dots , f_n)^k$ .

In fact, setting $J \mathcal {O}_Y = \mathcal {O}_Y(-E)$ , we even have that the canonical map

$$\begin{align*}{\mathbb R}\Gamma(Y, \mathcal{O}_Y(-(n+k-1)E)) \to L^k(\underline{f}) \otimes^{{\mathbb L}} {\mathbb R}\Gamma(Y, \mathcal{O}_Y) \end{align*}$$

is zero in the derived category.

The point is that if $r \in J^k$ , or equivalently if $r \in R$ maps to zero in $R/J^k$ , then r also maps to zero in $H_0( L^k(\underline {f}) \otimes ^{{\mathbb L}} {\mathbb R} \Gamma (Y, \mathcal {O}_{Y}))$ as we discuss below. But it turns out that mapping to zero in $H_0( L^k(\underline {f}) \otimes ^{{\mathbb L}} {\mathbb R} \Gamma (Y, \mathcal {O}_{Y}))$ is quite close to, and sometimes equivalent to, actually being in $J^k$ . An interesting aspect of the proof of this result is that it does not use any vanishing theorems or Cohen-Macaulayness, unlike many proofs of Briançon-Skoda-type results. Let us briefly explain the complex $L^k(\underline {f})$ . If $f_1, \dots , f_n$ is a regular sequence on R, then $L^k(\underline {f})$ resolves $R/J^k$ , that is, it is isomorphic to $\mathbb{Z}[x_1, \dots , x_n]/(x_1, \dots , x_n)^k \otimes ^{{\mathbb L}}_{\mathbb{Z}[x_1, \dots , x_n]} R$ in the derived category where $x_i \mapsto f_i$ . In general, $L^k(\underline {f})$ is isomorphic to a specialization of the Eagon-Northcott complex. In particular, for $k = 1$ , it is simply the Koszul complex. See Section 2.2 and [Reference Eagon and NorthcottEN62, Reference Buchsbaum and EisenbudBE75, Reference SrinivasanSri86, Reference SrinivasanSri89] for more details about this complex.

A Noetherian reduced ring R is said to be a birational derived splinter if $R \to {\mathbb R} \Gamma (Y, \mathcal {O}_Y)$ splits for all proper birational maps $\pi : Y \to \mathrm {Spec} \, R$ . That is, if there exists $\varphi : {\mathbb R} \Gamma (Y, \mathcal {O}_Y) \to R$ such that $ R \to {\mathbb R} \Gamma (Y, \mathcal {O}_Y) \xrightarrow {\varphi } R $ is an isomorphism, see [Reference LyuLyu22, Reference BhattBha12, Reference KovácsKov00, Reference De Deyn, Lank, Manali-Rahul and VenkateshDLMV25]. We observe below in Lemma 3.9 that every pseudo-rational local ring is a birational derived splinter via an argument of Kovács (the converse holds in the Cohen-Macaulay analytically unramified case). If R is a birational derived splinter, we see that

$$\begin{align*}\ker\Big(R \to H_0(L^k({\underline f}) \otimes^{{\mathbb L}} {\mathbb R} \Gamma(Y, \mathcal{O}_Y))\Big) \subseteq \ker\Big(R \to H_0(L^k({\underline f}) \otimes^{{\mathbb L}} R)\Big) = J^k. \end{align*}$$

As a consequence we obtain the following result.

Corollary A (Theorem 3.12)

Suppose R is a birational derived splinter (e.g., R is pseudo-rational, or R is excellent and one of F-rational, BCM-rational or $+$ -rational, see Remark 3.10), then

$$\begin{align*}\overline{J^{n+k-1}} \subseteq J^k \end{align*}$$

for any n-generated ideal J. The result also holds for any ideal which locally has analytic spread $\leq n$ .

This recovers the main result of [Reference Aberbach and HunekeAH01] and proves its analog in mixed characteristic. It provides simple proofs of the main results in [Reference Lipman and SathayeLS81, Reference Lipman and TeissierLT81] and generalizes them to pseudo-rational singularities and beyond.

More generally, suppose R is a Noetherian integral domain and $Y \to \mathrm {Spec} \, R$ is an alteration with Y integral and nonsingular. After picking a map $\psi : {\mathbb R}\Gamma (Y, \mathcal {O}_Y) \to R$ such that $R \to {\mathbb R}\Gamma (Y, \mathcal {O}_Y) \xrightarrow {\psi } R$ is multiplication by $c \neq 0$ (which always exists), our method combined with the fact that Y is a global birational derived splinter shows that $c \overline {J^{n+k-1}} \subseteq J^k$ independent of J. Having such a uniform c was exactly the missing piece needed to prove the uniform Artin-Rees theorem for excellent rings of finite dimension or to prove the uniform Briançon-Skoda theorem for excellent reduced rings of finite dimension. While we do not know the existence of regular alterations in this level of generality, Gabber’s weak local uniformization ([Reference Illusie, Laszlo and OrgogozoILO14, Exposé VII, Theorem 1.1]) provides a local variant of regular alterations. Utilizing Gabber’s result together with an intricate construction of nonsingular diagrams of schemes and derived splitting techniques, we obtain a $c\neq 0$ such that $c \overline {J^{n+k-1}} \subseteq J^k$ independent of J, see Theorem 5.3. Then, combining Huneke’s work [Reference HunekeHun92] with [Reference ZhouZho07, Reference LyuLyu25], we obtain the following.

Main Application (Corollary 5.6, Corollary 5.4, cf. [Reference HunekeHun92, Conjectures 1.3 and 1.4])

Suppose R is a quasi-excellent Noetherian ring of finite dimension and $N \subseteq M$ are finitely generated R-modules. Then there exists an integer $\ell $ depending on N and M such that for all ideals $I\subseteq R$ and all $n \geq \ell $ , we have that

$$\begin{align*}I^n M \cap N \subseteq I^{n-\ell} M. \end{align*}$$

If, in addition, R is reduced, then there exists an integer k depending only on R so that for all ideals $I\subseteq R$ and all $n \geq k$ , we have

$$ \begin{align*}\overline{I^n}\subseteq I^{n-k}.\end{align*} $$

Returning to our more precise Briançon-Skoda theorem, a slight variation of our method also yields the following result.

Corollary B (Theorem 3.18)

If R is Du Bois, or lim-perfectoid pure (e.g., F-pure), or Cohen-Macaulay and lim-perfectoid injective (e.g., Cohen-Macaulay and F-injective), then

$$\begin{align*}\overline{J^{n+k}} \subseteq J^k \end{align*}$$

for any n-generated ideal J. The result also holds for any ideal which locally has analytic spread $\leq n$ .

In fact, our Theorem 3.18 works for any reduced blowup-square splinter, a characteristic-free weakening of Du Bois singularities, so it generalizes [Reference Huneke and WatanabeHW15, Reference Wheeler and ZhangWZ25] and extends them to mixed characteristic. In a related direction, we also obtain in Corollary 3.20 that $J_{\operatorname {\mathrm {{perfd}}}}\overline {J^{n+k-1}} \subseteq J^k$ for an n-generated ideal J in any perfectoid (so not Noetherian) ring.

Finally, our result also implies and unifies several ideal-closure versions of the Briançon-Skoda theorem. Via the canonical map ${L^k(\underline {f}) \to R/J^{k}}$ , we have $ \overline {J^{n+k-1}} \subseteq \ker \Big (R \to H_0( R/J^k \otimes ^{{\mathbb L}} {\mathbb R} \Gamma (Y, \mathcal {O}_Y)) \Big ).$ If R is Noetherian and reduced, consider the union over all proper birational maps $Y \to \mathrm {Spec} \, R$

$$\begin{align*}(J^k)^{\operatorname{\mathrm{Bir}}} := \bigcup_{Y \to \mathrm{Spec} \, R}\ker \Big(R \to H_0( R/J^k \otimes^{{\mathbb L}} {\mathbb R} \Gamma(Y, \mathcal{O}_Y)) \Big) \end{align*}$$

which we call the birational preclosure of $J^k$ . We view it as an analog of the $+$ -closure of an ideal but in the birational direction. In positive (respectively mixed) characteristic, $R \to {\mathbb R}\Gamma (Y, \mathcal {O}_Y)$ factors the map $R \to R^+$ (respectively $R \to \widehat {R^+}$ ), see [Reference BhattBha12, Reference BhattBha20, Reference Hochster and HunekeHH92, Reference Huneke and LyubeznikHL07]. Thus we immediately obtain the following.

Corollary C (Proposition 4.10, Proposition 4.11)

Suppose R is a Noetherian reduced ring with p in its Jacobson radical and I and $J = (f_1, \dots , f_n)$ are ideals. Then $I^{\operatorname {\mathrm {Bir}}} \subseteq I \widehat {R^+} \cap R$ and so $\overline {J^{n+k-1}} \subseteq (J^k \widehat {R^+}) \cap R$ , recovering [Reference Hochster and HunekeHH95, Theorem 7.1] in characteristic p and [Reference Rodríguez-Villalobos and SchwedeRS24, Corollary 5.1] in mixed characteristic. These then imply that $\overline {J^{n+k-1}} \subseteq (J^k)^*$ of [Reference Hochster and HunekeHH90, Theorem 5.4] and that $\overline {J^{n+k-1}} \subseteq (J^k)^{\mathrm {ep}}$ of [Reference HeitmannHei01, Theorem 4.2].

1.1 History of this paper and strategy of the proof

This paper came out of [Reference Epstein, McDonald, Rebecca and SchwedeEMRS25] as an attempt to find a Briançon-Skoda theorem for the Hironaka-closure in characteristic zero. The starting point was [Reference Lipman and TeissierLT81], where Lipman-Teissier showed that if $h \in \overline {J^n}$ for $J = (f_1, \dots , f_n)$ , then the Čech class $[{h \over f_1 \cdots f_n}]$ is zero in $H^n_J({\mathbb R}\Gamma (Y, \mathcal {O}_Y))$ where Y is the normalized blowup of J. As Koszul homology limits to local cohomology, we began by trying to show the stronger statement that $h \mapsto 0 \in H_0\big (\text {Kos}(\underline {f}, {\mathbb R} \Gamma (Y, \mathcal {O}_Y))\big )$ ; indeed for small values of n this is easy to see quite explicitly, and we were able to prove it in general. Replacing the Koszul complex with a Buchsbaum-Eisenbud complex (or equivalently an Eagon-Northcott complex), we obtained our main result by a direct computation of the double complex $L^k(\underline {f}) \otimes {\mathbb R}\Gamma (Y, \mathcal {O}_Y)$ . Finally, we discovered our less computational proof of the main theorem, also compare with [Reference Lazarsfeld and LeeLL07, Proof of Theorem B]. The proof of the main application came after we realized that our result yielded a short proof that $\mathrm {T}(R) \neq 0$ assuming the existence of regular alterations.

2 Integral closure and the Buchsbaum-Eisenbud complex

In this section, we show our main result. We begin with some preliminaries on Čech covers.

2.1 Partially normalized blowups and Čech covers

We begin by discussing partially normalized blowups of $J \subseteq R$ . For more information, we refer the reader to [Sta25, Tag 052P] and [Sta25, Tag 01OF]. For those most interested in the geometric case, what follows below becomes easier to contemplate if R is a domain and so we invite the reader to think in that case if they prefer. Suppose that $J = (f_1, \dots , f_n)$ . Then the blowup of J, $Y \to \mathrm {Spec} \, R$ is covered by affine charts corresponding to each $f_i$ , $Y_i := \mathrm {Spec} \, R[{f_1 \over f_i}, \dots , {f_n \over f_i}]$ where here we view $R_i := R[{f_1 \over f_i}, \dots , {f_n \over f_i}]$ as a subring of $R[{1 \over f_i}]$ (in particular, if $g f_i = 0$ , then g becomes zero in this ring). We remind the reader of the following.

(a) The blowups of J and $J^m$ agree for any integer $m> 0$ . This follows either via the universal property of blowups or direct computation. In particular, for the blowup of $J^m$ , one only needs the charts $Y_i$ which correspond to $f_i^m \in J^m$ .
(b) Suppose $h \in \overline {J}$ with associated ideal-integral closure relation $h^d + a_1 h^{d-1} + \dots + a_d = 0$ with $a_i \in J^i$ , then we see that $h/f_i$ satisfies the ring-integral closure relation
$$\begin{align*}\left({h \over f_i}\right)^d + {a_1 \over f_i} \left({h \over f_i}\right)^{d-1} + \dots + {a_d \over f_i^d} \in R_i. \end{align*}$$

Furthermore, if we blowup $(J, h)$ , the chart corresponding to inverting h is redundant. Indeed, in the Rees algebra $R[(J, h)t]$ (with t a dummy variable) we see, by utilizing the original integral equation for h, that a homogeneous prime containing $ft$ for all $f \in J$ must contain $ht$ . Hence we see that the blowup of $(J, h)$ is integral over the blowup of J. Taking a limit, the blowup of $\overline {J^{n+k-1}}$ has an integral map to the blowup of J.

In what follows, we set $Y \to \mathrm {Spec} \, R$ to be the blowup of $\overline {J^{n+k-1}}$ with the affine cover whose charts $Y_i$ correspond to $f_i$ as above. Since $Y \to \mathrm {Spec} \, R$ factors through the blowup of J, we have that $J \mathcal {O}_Y = \mathcal {O}_Y(-E)$ where E is an effective Cartier divisor (E is not necessarily a Weil divisor in our generality as Y may not be normal). Observe also that $\overline {J^{n+k-1}} \mathcal {O}_Y = \mathcal {O}_Y(-(n+k-1)E)$ .

2.2 The Buchsbaum-Eisenbud complex

In this subsection, we provide some background on the Buchsbaum-Eisenbud complex, first introduced in [Reference Buchsbaum and EisenbudBE75]. It is isomorphic to an appropriate Eagon-Northcott complex [Reference Eagon and NorthcottEN62]. We largely follow the exposition in [Reference SrinivasanSri89, Section 1].

Let $J=(f_1,\dots ,f_n)$ be an ideal and let $F\to R$ be a map of free modules where F has basis $e_1,\dots ,e_n$ and $e_i\mapsto f_i$ . The Buchsbaum-Eisenbud complex for $J^k$ is then the complex

$$\begin{align*}0\to L_n^k(F)\to L_{n-1}^k(F)\to\dots\to L_2^k(F)\to L_1^k(F)\to R\to0\end{align*}$$

where $L_i^k(F)$ is the image of the natural map

$$\begin{align*}\bigwedge^iF\otimes_RS_{k-1}F\to\bigwedge^{i-1}F\otimes_RS_kF.\end{align*}$$

It easily follows that the $0$ th homology of $L^k(\underline {f})$ is $R/J^k$ . Indeed, if $f_1,\dots , f_n$ form a regular sequence, then $L^k(\underline {f})$ is a free resolution of $R/J^k$ .

We now globalize this perspective to the blowup of our ideal. Let $Y\to \mathrm {Spec} \, R$ be a (partially) normalized blowup of $J=(f_1,\dots ,f_n)$ as above and recall that $J\mathcal {O}_Y=\mathcal {O}_Y(-E)$ . It is standard to create an exact Koszul-type complex of locally free sheaves on such a blowup (see for instance [Reference LazarsfeldLaz04, Proof of 9.6.31]); we explain the generalization of this to the Buchsbaum-Eisenbud complex or Eagon-Northcott complex. Let $\{Y_i\}_{i=1}^n$ be the standard affine charts and consider $L^k(R_j)$ , the Buchsbaum-Eisenbud complex associated to the kth power of the (unit) ideal $(\frac {f_1}{f_j},\dots ,\frac {f_n}{f_j})\subseteq R_j$ :

$$ \begin{align*}0 \to R_j^{b_n} \xrightarrow{\partial_n} R_j^{b_{n-1}} \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\partial_3} R_j^{b_2} \xrightarrow{\partial_2} R_j^{b_1}\xrightarrow{\partial_1} R_j \to 0.\end{align*} $$

Note that $\partial _1\colon R_j^{b_1}\to R_j$ has as its entries the degree k monomials in $\frac {f_1}{f_j},\dots ,\frac {f_n}{f_j}$ (which includes $\frac {f_j}{f_j}=1$ ) while the remaining differentials are matrices whose entries are $\pm \frac {f_i}{f_j}$ . Furthermore, this complex is exact as its homologies are annihilated by the unit ideal. We next consider the following map of complexes:

Commutative diagram of two exact sequences. Top row: 0 to R sub j super b sub n to R sub j super b sub n-1 to R sub j super b sub 1 to R sub j to 0. Bottom row: 0 to f sub j super -(n-1) R sub j super b sub n to f sub j super k R sub j to 0.

By design, this map is an isomorphism. Furthermore, the differentials of the bottom complex are simply matrices whose entries are $\pm f_i$ (except for $f_j^k\partial _1$ whose entries are the degree k monomials in $f_1,\dots ,f_n$ ) and thus its differentials are independent of j. Therefore, we can glue these complexes on the charts $\{Y_i\}_{i=1}^n$ together to obtain the following exact complex:

$$\begin{align*}0 \to \mathcal{O}_Y((n-1)E)^{\oplus b_n}\to \mathcal{O}_Y((n-2)E)^{\oplus b_{n-1}}\to\cdots\to\mathcal{O}_Y^{\oplus b_1}\to \mathcal{O}_Y(-kE) \to 0. \end{align*}$$

Twisting by $\mathcal {O}_Y(-(n-1)E)$ , we obtain an exact complex

$$\begin{align*}\text{BE}_2\colon \hspace{0.5em} 0 \to \mathcal{O}_Y^{\oplus b_n}\to \mathcal{O}_Y(-E)^{\oplus b_{n-1}}\to\cdots\to\mathcal{O}_Y(-(n-1)E)^{\oplus b_1}\to \mathcal{O}_Y(-(n+k-1)E) \to 0 \end{align*}$$

which can be viewed as a subcomplex of the Buchsbaum-Eisenbud complex for $(f_1,\dots ,f_n)^k$ on Y:

$$\begin{align*}\text{BE}_1\colon\hspace{0.5em} 0 \to \mathcal{O}_Y^{\oplus b_n}\to \mathcal{O}_Y^{\oplus b_{n-1}}\to\cdots\to\mathcal{O}_Y^{\oplus b_1}\to \mathcal{O}_Y\to 0. \end{align*}$$

Remark 2.1. Note that the Buchsbaum-Eisenbud complex is isomorphic to the so-called Eagon-Northcott complex by [Reference Buchsbaum and EisenbudBE75, Cor. 3.2]. We chose to work with the Buchsbaum-Eisenbud complex here, as it gives the obvious generating set for $(f_1,\dots ,f_n)^k$ and is generally easier to describe than the Eagon-Northcott complex. Readers familiar with the Eagon-Northcott complex can also check that there is a map from an exact Eagon-Northcott complex $\text {EN}_2$ on Y to the pullback $\text {EN}_1$ of an Eagon-Northcott complex on $\mathrm {Spec} \, R$ to Y, where

$$ \begin{align*} \text{EN}_2\colon \hspace{1em} & 0\to \bigwedge^{n+k-1}\mathcal{O}_Y^{\oplus n+k-1} \otimes (\operatorname{\mathrm{Sym}}^{n-1}\mathcal{O}_Y(-E)^{\oplus k})^*\otimes \mathcal{O}_Y(-(n-1)E) \to \cdots \\ & \to \bigwedge^{k+1}\mathcal{O}_Y^{n+k-1}\otimes (\operatorname{\mathrm{Sym}}^1\mathcal{O}_Y(-E)^{\oplus k})^* \otimes \mathcal{O}_Y(-(n-1)E)\\ & \to \bigwedge^{k}\mathcal{O}_Y^{n+k-1} \otimes \mathcal{O}_Y(-(n-1)E) \\ & \to \bigwedge^k\mathcal{O}_Y(-E)^{\oplus k} \otimes \mathcal{O}_Y(-(n-1)E) \to 0 \end{align*} $$

StartLayout 1st Row 1st Column EN Subscript 2 Baseline colon 2nd Column 0 right arrow and Overscript n plus k minus 1 Endscripts script upper O Subscript upper Y Superscript circled plus n plus k minus 1 Baseline circled times left parenthesis upper S y m Superscript n minus 1 Baseline script upper O Subscript upper Y Baseline left parenthesis negative upper E right parenthesis Superscript circled plus k Baseline right parenthesis Superscript asterisk Baseline circled times script upper O Subscript upper Y Baseline left parenthesis minus left parenthesis n minus 1 right parenthesis upper E right parenthesis right arrow midline horizontal ellipsis 2nd Row 1st Column Blank 2nd Column right arrow and Overscript k plus 1 Endscripts script upper O Subscript upper Y Superscript n plus k minus 1 Baseline circled times left parenthesis upper S y m Superscript 1 Baseline script upper O Subscript upper Y Baseline left parenthesis negative upper E right parenthesis Superscript circled plus k Baseline right parenthesis Superscript asterisk Baseline circled times script upper O Subscript upper Y Baseline left parenthesis minus left parenthesis n minus 1 right parenthesis upper E right parenthesis 3rd Row 1st Column Blank 2nd Column right arrow and Overscript k Endscripts script upper O Subscript upper Y Superscript n plus k minus 1 Baseline circled times script upper O Subscript upper Y Baseline left parenthesis minus left parenthesis n minus 1 right parenthesis upper E right parenthesis 4th Row 1st Column Blank 2nd Column right arrow and Overscript k Endscripts script upper O Subscript upper Y Baseline left parenthesis negative upper E right parenthesis Superscript circled plus k Baseline circled times script upper O Subscript upper Y Baseline left parenthesis minus left parenthesis n minus 1 right parenthesis upper E right parenthesis right arrow 0 EndLayout

while

$$ \begin{align*} \text{EN}_1\colon \hspace{0.5em} & 0 \to \bigwedge^{n+k-1}\mathcal{O}_Y^{\oplus n+k-1} \otimes (\operatorname{\mathrm{Sym}}^{n-1}\mathcal{O}_Y^{\oplus k})^* \to \cdots\\ & \to \bigwedge^{k+1}\mathcal{O}_Y^{n+k-1}\otimes (\operatorname{\mathrm{Sym}}^1\mathcal{O}_Y^{\oplus k})^* \to \bigwedge^{k}\mathcal{O}_Y^{n+k-1} \to \bigwedge^k\mathcal{O}_Y^{\oplus k} \to 0. \end{align*} $$

StartLayout 1st Row 1st Column EN Subscript 1 Baseline colon 2nd Column 0 right arrow and Overscript n plus k minus 1 Endscripts script upper O Subscript upper Y Superscript circled plus n plus k minus 1 Baseline circled times left parenthesis upper S y m Superscript n minus 1 Baseline script upper O Subscript upper Y Superscript circled plus k Baseline right parenthesis Superscript asterisk Baseline right arrow midline horizontal ellipsis 2nd Row 1st Column Blank 2nd Column right arrow and Overscript k plus 1 Endscripts script upper O Subscript upper Y Superscript n plus k minus 1 Baseline circled times left parenthesis upper S y m Superscript 1 Baseline script upper O Subscript upper Y Superscript circled plus k Baseline right parenthesis Superscript asterisk Baseline right arrow and Overscript k Endscripts script upper O Subscript upper Y Superscript n plus k minus 1 Baseline right arrow and Overscript k Endscripts script upper O Subscript upper Y Superscript circled plus k Baseline right arrow 0 period EndLayout

These two complexes are both constructed from the $k\times (n+k-1)$ matrix:

$$\begin{align*}\begin{bmatrix} f_1 & f_2 & \dots & f_n & 0 & \dots & \dots & 0\\ 0 & f_1 & f_2 & \dots & f_n & 0 &\dots & 0 \\ 0 & 0 & f_1 & \dots & \dots & f_n & \dots, & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & \dots & \dots & \dots & f_n \end{bmatrix}. \end{align*}$$

Start 5 By 8 Matrix 1st Row 1st Column f 1 2nd Column f 2 3rd Column ellipsis 4th Column f Subscript n Baseline 5th Column 0 6th Column ellipsis 7th Column ellipsis 8th Column 0 2nd Row 1st Column 0 2nd Column f 1 3rd Column f 2 4th Column ellipsis 5th Column f Subscript n Baseline 6th Column 0 7th Column ellipsis 8th Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column f 1 4th Column ellipsis 5th Column ellipsis 6th Column f Subscript n Baseline 7th Column ellipsis comma 8th Column 0 4th Row 1st Column vertical ellipsis 2nd Column vertical ellipsis 3rd Column vertical ellipsis 4th Column down right diagonal ellipsis 5th Column vertical ellipsis 6th Column down right diagonal ellipsis 7th Column down right diagonal ellipsis 8th Column vertical ellipsis 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column ellipsis 5th Column ellipsis 6th Column ellipsis 7th Column ellipsis 8th Column f Subscript n Baseline EndMatrix period

2.3 The main result

In this subsection, we prove our derived version of the Briançon-Skoda theorem. The proof is remarkably simple and short.

Theorem 2.2. Suppose R is a ring and $J = (f_1, \dots , f_n)\subseteq R$ is an ideal. Suppose that $\pi : Y \to \mathrm {Spec} \, R$ is the blowup of $\overline {J^{n+k-1}}$ (or any map that dominates it) with $J\mathcal {O}_Y=\mathcal {O}_Y(-E)$ for E an effective Cartier divisor. Then the canonical map

$$\begin{align*}{\mathbb R}\Gamma(Y, \mathcal{O}_Y(-(n+k-1)E)) \to L^k(\underline{f}) \otimes^{{\mathbb L}} {\mathbb R}\Gamma(Y, \mathcal{O}_Y) \end{align*}$$

is the zero map in the derived category. In fact, even $\mathcal {O}_Y(-(n+k-1)E) \to L^k(\underline {f}) \otimes \mathcal {O}_Y$ is zero. In particular, by taking zeroth cohomology, the map

$$\begin{align*}\overline{J^{n+k-1}} \to H_0(L^k(\underline{f})\otimes^{\mathbb{L}} {\mathbb R}\Gamma(Y,\mathcal{O}_Y)) \end{align*}$$

is zero.

Proof. By viewing $\mathcal {O}_Y(-(n+k-1)E)$ as a complex in degree zero, we have a natural map $\mathcal {O}_Y(-(n+k-1)E) \to \text {BE}_2$ and hence a natural composition map

$$\begin{align*}\mathcal{O}_Y(-(n+k-1)E) \to \text{BE}_2 \to \text{BE}_1. \end{align*}$$

SinceFootnote ⁴ $\text {BE}_2$ is exact and hence zero in the derived category, $\mathcal {O}_Y(-(n+k-1)E) \to \text {BE}_1 = L^k(\underline {f})\otimes \mathcal {O}_Y$ is zero in the derived category. Now simply apply the derived functor ${\mathbb R}\Gamma (Y, -)$ and use the projection formula to obtain the conclusion. The final assertion follows by taking the zeroth cohomology and noticing that $\overline {J^{n+k-1}}\subseteq \Gamma (Y, \mathcal {O}_Y(-(n+k-1)E))$ .

Remark 2.3. There is another short argument explaining Theorem 2.2. For any invertible ideal sheaf $\mathcal {I}\subseteq \mathcal {O}_Y$ that is globally generated by n elements $(f_1,\dots ,f_n)$ , the $\mathcal {O}_Y$ -module structure on the $\mathcal {O}_Y$ -algebra $\text {Kos}(f_1,\dots ,f_n; \mathcal {O}_Y)$ factors over $\mathcal {O}_Y/\mathcal {I}^n$ : it has n cohomology groups (that sit in cohomological degrees $[-(n-1),0]$ ), each of which is annihilated by $\mathcal {I}$ , and thus $\mathcal {I}^{n}$ annihilates $\text {Kos}(f_1,\dots ,f_n; \mathcal {O}_Y)$ in $D(Y)$ by [Reference BhattBha12, Lemma 3.2]. In general, by comparing with the exact complex $\text {BE}_2$ one sees that $H^{i}(\text {BE}_1)$ is annihilated by $\mathcal {I}$ for $-(n-1)\leq i\leq -1$ while $H^0(\text {BE}_1)$ is annihilated by $\mathcal {I}^k$ . Thus $\mathcal {I}^{n+k-1}$ annihilates $\text {BE}_1$ in $D(Y)$ by [Reference BhattBha12, Lemma 3.2]. We would like to thank Bhargav Bhatt for pointing this out to us.

Remark 2.4. The statement and proof of Theorem 2.2 also work verbatim if $\overline {J^{n+k-1}}$ is replaced, wherever it occurs, by any ideal I such that $J^{n+k-1} \subseteq I \subseteq \overline {J^{n+k-1}}$ .

3 Generalizations of rational and Du Bois singularities

Suppose $\pi : Y \to X$ is a map of reduced Noetherian schemes. We say $\pi $ is birational if $\pi $ induces a bijection between the generic points of X and Y and furthermore induces an isomorphism between the residue fields at those generic points, see [Sta25, Tag 01RO]. If $\pi $ is of finite type, this is equivalent to the condition that $\pi $ induces a bijection between the irreducible components of Y and of X, and $\pi $ induces an isomorphism on a dense open set on each such irreducible component, see [Sta25, Tag 0BAC].

We recall the definition of pseudo-rational rings, noting that regular rings are pseudo-rational [Reference Lipman and TeissierLT81, Section 4].

Definition 3.1 [Reference Lipman and TeissierLT81, Section 2]

Suppose $(R, \mathfrak {m})$ is a d-dimensional Noetherian local ring. We say that R has pseudo-rational singularities if the following conditions hold.

(a) R is normal.
(b) R is Cohen-Macaulay.
(c) The $\mathfrak {m}$ -adic completion $\widehat {R}$ is reduced (that is, R is analytically unramified).
(d) For every proper birational map $\pi : Y \to \mathrm {Spec} \, R$ from a normal and integral Y, the induced
$$\begin{align*}H^d_{\mathfrak{m}}(R) \to H^d_{\mathfrak{m}}({\mathbb R} \Gamma(Y, \mathcal{O}_Y)) \end{align*}$$

is injective. If R has a dualizing module $\omega _R$ , this just means that $\Gamma (Y, \omega _Y) \to \omega _R$ is surjective.

We say that a Noetherian scheme is pseudo-rational if its stalks are pseudo-rational.

The earlier conditions of the definition imply that there are many normal Y with $\pi : Y \to \mathrm {Spec} \, R$ proper birational. In fact, for each proper birational $Y \to \mathrm {Spec} \, R$ with Y integral, the normalization map $Y^{\mathrm {N}} \to Y$ is finite, see the discussion [Reference Lipman and TeissierLT81, Pages 102, 103].

Our next goal is to study a weakening of the pseudo-rational hypothesis. For that we need a definition and some basic facts about pure maps in the derived category.

Definition 3.2 [Reference ChristensenChr98],[Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMP+24, Proposition 2.10]

A map $M \to N$ in $D(R)$ with cone Q is pure if for each perfect complex P with a map $P \to Q$ , the composition $P \to M[1]$ is zero. In other words, this means that $Q \to M[1]$ is phantom in the sense of [Reference ChristensenChr98], see [Reference KrauseKra22, Chapter 5].

Equivalently, $M \to N$ is pure if $M \to N$ can be written as a filtered colimit of split maps $M \to N_i$ (in the derived $\infty $ -category).

Lemma 3.3 (cf. [Reference LyuLyu22, Proposition 3.10])

Suppose R is a ring and $R \to S$ is faithfully flat. Suppose $M \to N$ is a map in $D(R)$ such that the base change $M \otimes _R S=: M_S \to N_S := N \otimes _R S$ is pure. Then $M \to N$ is also pure.

Proof. Let Q denote the cone of $M \to N$ and let $P \to Q$ be a map from a perfect complex to Q. Our assumption implies that the base change $Q_S \to P_S \to M_S[1]$ is zero. But $\operatorname {\mathrm {Hom}}_{D(R)}(P, Q) \to \operatorname {\mathrm {Hom}}_{D(S)}(P_S, Q_S)$ is injective since P is perfect [Sta25, Tag 0A6A] and so the result follows.

The following result generalizes the fact that a map of modules $f : M \to N$ with $N/f(M)$ finitely presented is pure if and only if it is split injective ([Reference Hochster and RobertsHR76, Corollary 5.2], [Sta25, Tag 058K]).

Lemma 3.4. Suppose R is Noetherian, $M \to N \to C \xrightarrow {+1}$ is a triangle in $D(R)$ with $M\in D^+(R)$ and $C \in D^-_{\mathrm {coh}}(R)$ . Then $M \to N$ is pure if and only if $M \to N$ has a left inverse in $D(R)$ .

Proof. Obviously if the map is split, then it is pure. Srikanth Iyengar pointed out that the converse can be deduced from [Reference KrauseKra22, Proposition 5.2.8] but we provide a proof for the reader’s convenience. First note that since $M\in D^+(R)$ , there exists n so that $\operatorname {\mathrm {Hom}}_{D(R)}(K, M)=\operatorname {\mathrm {Hom}}_{D(R)}(\tau ^{\geq -n}K, M)$ for all $K\in D(R)$ . Since $C \in D^-_{\mathrm {coh}}(R)$ , we can write C as a colimit of perfect complexes $C_i$ so that $\tau ^{\geq -n}C_i\simeq \tau ^{\geq -n}C$ for all i. To show $M\to N$ is split in $D(R)$ , we need to show that $\mathrm {Ext}^1(C, M)=H^1({\mathbb R}\lim {\mathbb R}\operatorname {\mathrm {Hom}}(C_i, M))=0$ . By the Milnor exact sequence [Sta25, Tag 0CQE], we have

$$ \begin{align*}0\to {\lim}^1\operatorname{\mathrm{Hom}}(C_i, M)\to H^1({\mathbb R}\lim{\mathbb R}\operatorname{\mathrm{Hom}}(C_i, M)) \to \lim\mathrm{Ext}^1(C_i, M) \to 0.\end{align*} $$

The right term above vanishes because $M\to N$ is pure and $C_i$ is perfect, and the left term above also vanishes since $\operatorname {\mathrm {Hom}}(C_i, M) = \operatorname {\mathrm {Hom}}(\tau ^{\geq -n}C_i, M)=\operatorname {\mathrm {Hom}}(\tau ^{\geq -n}C, M)$ and thus $\{\operatorname {\mathrm {Hom}}(C_i, M)\}$ is a constant system. Thus $\mathrm {Ext}^1(C, M)=0$ as we wanted.

Lyu coined the following term, also see [Reference KovácsKov00, Reference MurayamaMur25, Reference De Deyn, Lank, Manali-Rahul and VenkateshDLMV25].

Definition 3.5 [Reference LyuLyu22]

A Noetherian reduced ring R is a birational derived splinter if for each proper birational $\pi : Y \to \mathrm {Spec} \, R$ , we have that $R \to {\mathbb R} \Gamma (Y, \mathcal {O}_Y)$ has a left inverse. More generally, a Noetherian reduced scheme X is a global birational derived splinter if for each proper birational $\pi : Y \to X$ , we have that $\mathcal {O}_X \to {\mathbb R} \pi _* \mathcal {O}_Y$ has a left inverse.

Remark 3.6. By Chow’s lemma, to show that R is a birational derived splinter, it suffices to consider projective and birational $\pi $ , hence simply consider blowups. See [Reference LiuLiu02, Chapter 8, Theorem 8.1.24].

Remark 3.7. Birational derived splinters are easily seen to be normal. Indeed, if $R \to S$ is a finite birational extension, then $R \to S$ is not pure (equivalently split), unless $S = R$ (as the conductor of the map is the image of $\operatorname {\mathrm {Hom}}_R(S, R) \to R$ ). Hence, restricting birational derived splinters to domains is harmless since if R has minimal primes $Q_1, \dots , Q_n$ , then $R \to \prod R/Q_i$ is a finite extension.

Remark 3.8. In what follows, it will often be helpful to replace our local ring $(R,\mathfrak {m})$ with its strict Henselization $R^{\operatorname {\mathrm {{sh}}}}$ in order to guarantee that R has an infinite residue field. As such, we want to ensure that if R is a birational derived splinter (or a reduced blowup-square splinter, see Definition 3.15) then so is $R^{\operatorname {\mathrm {{sh}}}}$ . This follows because proper birational maps to $\mathrm {Spec} \, R^{\operatorname {\mathrm {{sh}}}}$ can be dominated by the base change of a proper birational map to $\mathrm {Spec} \, R$ by [Reference LyuLyu22, Theorem 5.1]. Alternatively, a similar reduction to the infinite residue field case can be obtained with $R[t]_{\mathfrak {m}[t]}$ , see for instance [Reference Lipman and TeissierLT81, Example (c) before Theorem 2.1].

We believe the following is known to experts; indeed, the argument can be found in the proof of [Reference KovácsKov00, Lemma 2]. In the case that X is regular, it also follows from [Reference Lank and VenkateshLV24, Lemma 3.16].

Lemma 3.9. Suppose $(R, \mathfrak {m})$ is a pseudo-rational local ring of dimension d, then R is a birational derived splinter. Even more, if X is a pseudo-rational scheme with a canonical sheaf, then X is a global birational derived splinter.

Proof. Suppose $\pi : Y \to \mathrm {Spec} \, R$ is a proper birational map with Y normal. We work with the completion $\widehat {R}$ to gain access to a canonical module. By Lemma 3.3 and Lemma 3.4, it suffices to show $ \widehat {R} \to \widehat {R} \otimes ^{{\mathbb L}} {\mathbb R}\Gamma (Y, \mathcal {O}_Y) \cong {\mathbb R} \Gamma (Y_{\widehat {R}}, \mathcal {O}_{Y_{\widehat {R}}}) $ is pure where $Y_{\widehat {R}} := Y \times _{\mathrm {Spec} \, R} \mathrm {Spec} \, \widehat {R}$ . We will show that $ \widehat {R} \to {\mathbb R} \Gamma (Y_{\widehat {R}}, \mathcal {O}_{Y_{\widehat {R}}})$ is split.

Now, $\widehat {R}$ is Cohen-Macaulay and has a normalized dualizing complex $\omega _{\widehat {R}}[d]$ . Since $H^d_{\mathfrak {m}}(R) \to H^d_{\mathfrak {m}}({\mathbb R} \Gamma (Y, \mathcal {O}_Y))$ injects, we see that the Matlis dual $ \Gamma (Y_{\widehat {R}}, \omega _{Y_{\widehat {R}}}) \to \omega _{\widehat {R}} $ surjects and so is an isomorphism (both sides are rank-1 torsion-free). This yields a composition

that one sees is an isomorphism (it is a map between S2-modules that is an isomorphism outside a set of codimension $\geq 2$ ). Applying Grothendieck duality produces a composition

$$\begin{align*}\widehat{R} \leftarrow {\mathbb R}\Gamma(Y_{\widehat{R}}, \mathcal{O}_{Y_{\widehat{R}}}) \leftarrow \widehat{R} \end{align*}$$

that is also an isomorphism. This proves that $\widehat {R} \to {\mathbb R} \Gamma (Y_{\widehat {R}}, \mathcal {O}_{Y_{\widehat {R}}})$ is split, as desired.

For the scheme case, since one may assume X is integral, we may assume that is a dualizing complex. Set . As above, we have a composition

which is the identity. Then dualize and argue as above.

Remark 3.10. Excellent local rings that are F-rational or BCM-rational (or even $+$ -rational: i.e., $(R,\mathfrak {m},k)$ is a Cohen-Macaulay local domain with $\mathrm {char}\,k> 0$ , such that $H^{\dim R}_{\mathfrak {m}}(R) \to H^{\dim R}_{\mathfrak {m}}(S)$ injects for all finite $R \subseteq S$ ) are birational derived splinters, as all are pseudo-rational [Reference SmithSmi97, Reference Ma and SchwedeMS21]. Regular rings are derived splinters in general [Reference BhattBha18], but the argument above that they are birational derived splinters is much simpler than the fact that finite maps $R \to S$ split in mixed characteristic [Reference AndréAnd18].

Remark 3.11. We note that excellent local Cohen-Macaulay birational derived splinters are pseudo-rational as condition (d) of Definition 3.1 is clear from the splinter condition. However, without the Cohen-Macaulay condition, this is not true.

Suppose $R \subseteq S$ is a pure inclusion of Noetherian domains and S is a birational derived splinter, then it is not difficult to see that R is a birational derived splinter as well, see [Reference LyuLyu22, Corollary 3.12]. But now by [Reference Ma and PolstraMP25, Example 8.6] (which utilizes the examples of [Reference KovácsKov18]), there exists such a split extension $R \subseteq S$ so that S is F-rational and hence pseudo-rational but R is not Cohen-Macaulay. In particular, in this example, R is a birational derived splinter but is not pseudo-rational.

3.1 Briançon-Skoda theorem for birational derived splinters

Our discussion so far yields the following result.

Theorem 3.12. Suppose that a Noetherian ring R is a birational derived splinter (for instance if R is pseudo-rational, or excellent and one of F-rational, BCM-rational or $+$ -rational). Suppose that $J \subseteq R$ is an ideal that locally has analytic spread $\leq n$ . Then for every integer $k> 0$

$$\begin{align*}\overline{J^{n+k-1}} \subseteq J^k. \end{align*}$$

Proof. We may assume that R is a normal local domain. Let Y denote the blowup of $\overline {J^{n+k-1}}$ . We may also assume that R is strictly Henselian (see [Reference Huneke and SwansonHS06, Proposition 1.6.2] and Remark 3.8, and note that blowups commute with flat base change) and so assume R has infinite residue field.

By [Reference Huneke and SwansonHS06, Proposition 8.3.7], J has a minimal reduction $J' = (f_1, \dots , f_n) \subseteq J$ whose powers have the same integral closures as the powers of J. Hence we may replace J by $J'$ to assume that J is n-generated. Now by Theorem 2.2 and using that $R \to {\mathbb R}\Gamma (Y, \mathcal {O}_Y)$ splits, we have that $\overline {J^{n+k-1}}$ maps to zero in $H_0(L^k(\underline {f})) \otimes R = R/J^k$ , that is, $\overline {J^{n+k-1}} \subseteq J^k$ .

Remark 3.13. Suppose that $Y \to \mathrm {Spec} \, R$ is the blowup of $\overline {J^{n+k-1}}$ and that there exists some map $\varphi : {\mathbb R}\Gamma (Y, \mathcal {O}_Y) \to R$ such that the composition $R \to {\mathbb R}\Gamma (Y, \mathcal {O}_Y) \xrightarrow {\varphi } R$ is multiplication by c (see [Reference Epstein, McDonald, Rebecca and SchwedeEMRS25, Definition 2.21] and [Reference LyuLyu22, Section 2.4]). The proof above shows that

$$\begin{align*}c \overline{J^{n+k-1}} \subseteq J^k. \end{align*}$$

For more discussion in this direction, see Section 5.

Remark 3.14. Suppose R is a normal domain essentially of finite type over $\mathbb{C}$ . If R has rational singularities, then we know from [Reference Aberbach and HunekeAH01] that the Briançon-Skoda theorem holds. In [Reference LiLi21], the author gave another proof of this fact in the analytic setting, and also showed that $ \overline {\mathcal {J}^{n+k-1}}\cdot \omega _R\subset \mathcal {J}^k\cdot \omega _R$ assuming that $\omega _R = \Gamma (X, \omega _X)$ for $X \to \mathrm {Spec} \, R$ a resolution of singularities (roughly, R is pseudo-rational without the Cohen-Macaulay hypothesis). It was also claimed that one only needed this weakening of rational singularities without the Cohen-Macaulay condition to deduce that $\overline {J^{n+k-1}} \subseteq J^k$ generally.

We believe this final statement is false in the algebraic category. Explicitly, let E denote the elliptic curve defined by $x^3+y^3+z^3 = 0$ in $\mathbb {P}^2_{\mathbb {C}}$ and set $X = E \times \mathbb {P}^1_{\mathbb {C}}$ Segre embedded in $\mathbb {P}^5_{\mathbb {C}}$ . Set R to be the affine cone over X. It is well-known that R is not Cohen-Macaulay (use the Künneth formula to see that $H^1(X, \mathcal {O}_X)\neq 0$ ). We explain why $\omega _R = {\mathbb R}\Gamma (Y, \omega _Y)$ where $Y \to \mathrm {Spec} \, R$ is the blowup of the origin. By [Reference UrbinatiUrb12, Theorem 4.1] and [Reference De Fernex and HaconDH09, Theorem 8.2] we see that $\omega _R \cdot \mathcal {O}_Y \subseteq \omega _Y$ . Hence we have a sequence of maps

$$\begin{align*}\omega_R \to \Gamma(Y, \omega_R \cdot \mathcal{O}_Y ) \to \Gamma(Y, \omega_Y) \to \omega_R. \end{align*}$$

But these are nonzero maps of rank-1 torsion-free modules and hence they are injective. Furthermore, these maps are the identity outside of the origin and hence the composition is an isomorphism as $\omega _R$ is reflexive. It follows that $\Gamma (Y, \omega _Y) \to \omega _R$ is an isomorphism.

Now, it is well-known that R is the Segre product $\mathbb {C}[x,y,z]/(x^3+y^3+z^3) \# \mathbb {C}[u,v]$ . Let $J = ((xu)^2, (zu)^2) \subseteq R$ and $J' = ((xu)^2, (xu)(zu),(zu)^2) \subseteq \overline {J}$ . Then we have $h := (xu)(yu)^2(zu) \in \overline {J'^2}=\overline {J^2}$ (use $h^3 \in J'^6$ for the first containment). On the other hand, if $h\in J$ , then after specializing $u=v=1$ , we would have $xy^2z\in (x^2,z^2)S$ where $S=\mathbb {C}[x,y,z]/(x^3+y^3+z^3)$ . But since $x, z$ is a regular sequence in S, this would imply $y^2\in (x,z)S$ which is a contradiction. Thus $h\notin J$ . We found this example with the help of Macaulay2 [Reference Grayson and StillmanGS].

3.2 A characteristic-free weakening of Du Bois singularities

Suppose R is reduced and $Y \to \mathrm {Spec} \, R$ is the blowup of an ideal $J \subseteq R$ . Consider the reduced blowup square

Commutative diagram where F maps to Y and V(square root of J). Y maps to Spec R. V(square root of J) maps to Spec R. Arrows point right and down.

with F the reduced preimage of $V(\sqrt {J})$ . We have the following triangle defining :

(1)

Definition 3.15. Suppose R is a Noetherian reduced ring. We say that R is a reduced blowup-square splinter if for each ideal $J\subseteq R$ and as above, the induced map has a left inverse.

Suppose R is essentially of finite typeFootnote ⁵ over a field of characteristic zero. Then, for instance by [Reference Du BoisDB81, Reference Guillén, Navarro Aznar, Pascual Gainza and PuertaGNPP88], we have functorial maps $R \to {\underline \Omega {}^0_{R}}, \mathcal {O}_Y \to {\underline \Omega {}^0_{Y}}, \mathcal {O}_F \to {\underline \Omega {}^0_{F}}, $ and $R/\sqrt {J} \to {\underline \Omega {}^0_{R/\sqrt {J}}}$ where ${\underline \Omega {}^0_{}}$ is the $0$ th-graded piece of the Deligne-Du Bois complex. Hence from the triangle (see [Reference Du BoisDB81, Proposition 3.9]) $ {\underline \Omega {}^0_{R}} \to {\mathbb R}\Gamma (Y, {\underline \Omega {}^0_{Y}}) \oplus {\underline \Omega {}^0_{R/\sqrt {J}}} \to {\mathbb R}\Gamma (Y, {\underline \Omega {}^0_{F}}) \xrightarrow {+1} $ we have a factorization

(2)

This factorization also follows from hyperresolution constructions [Reference Guillén, Navarro Aznar, Pascual Gainza and PuertaGNPP88] or from [Reference Huber and JörderHJ14].

Proposition 3.16. The following singularities are reduced blowup-square splinters.

(a) R is essentially of finite type over a field of characteristic zero and is Du Bois.
(b) R has p in its Jacobson radical and is lim-perfectoid pure (e.g., R is F-pure).
(c) R has p in its Jacobson radical and is Cohen-Macaulay and lim-perfectoid injective (e.g., R is Cohen-Macaulay and F-injective).

Proof. For (a), recall that R having Du Bois singularities means exactly that $R \to {\underline \Omega {}^0_{R}}$ is an isomorphism. Thus (2) gives a left inverse of .

For lim-perfectoid pure singularities, we have a factorization

see for instance [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMP+24, Lemma 5.3]. Thus by the definition of lim-perfectoid purity, $R\to R_{\operatorname {\mathrm {{perfd}}}}$ is pure and thus is pure in $D(R)$ . Hence the latter is split in $D(R)$ by Lemma 3.4 (note that in characteristic p, lim-perfectoid purity is precisely F-purity by [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMP+24, Remark 4.3]).

Now we assume R is Cohen-Macaulay and lim-perfectoid injective (by [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMP+24, Remark 4.3], in characteristic p this is equivalent to F-injective). By considering the factorization as above and the definition of lim-perfectoid injective, we have that is injective for each maximal ideal $\mathfrak {m}$ . After base changing to the $\mathfrak {m}$ -adic completion $\widehat {R}$ , we have that is injective. Hence by local duality, we have a surjection

where is Grothendieck duality and is a normalized dualizing complex.

Claim 3.17. The map is an injection.

Proof of claim

The strategy is the same as [Reference Kovács, Schwede and SmithKSS10, Proposition 4.10]. If we consider the triangle it suffices to show that . From the spectral sequence , it suffices to show that for every i.

Now, as R is weakly normal (see [Reference Datta and MurayamaDM24, Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMP+24]), by taking the zeroth cohomology of the sequence (1), we see that . In particular, for $i> 0$ and . From (1) we see that there is a triangle

It follows that for $i> 0$ . Finally, $\dim \operatorname {\mathrm {Supp}} H^i(Y, I_F) \leq d-i-1$ as $Y \to \mathrm {Spec} \, R$ has fiber dimension $\leq \dim (R_Q) -1$ at every localization $R_Q$ of R. This proves the claim.

Finally, we have a composition which is an isomorphism as in the pseudo-rational case:

Dualizing proves that splits for each maximal ideal $\mathfrak {m}$ . Using Lemma 3.3, Lemma 3.4 and [Sta25, Tag 0D0C], similar to [Reference LyuLyu22, 2.4, Proposition 3.5], completes the proof.

Our next result generalizes [Reference Huneke and WatanabeHW15, Theorem 3.2(2)] and several results in [Reference Wheeler and ZhangWZ25] and answers some questions therein.

Theorem 3.18. Suppose R is a reduced ring and $J = (f_1, \dots , f_n) \subseteq R$ is an ideal. Let $\pi : Y \to \mathrm {Spec} \, R$ denote the blowup of $I := \overline {J^{n+k-1}}$ with $J \mathcal {O}_Y = \mathcal {O}_Y(-E)$ so that $I \mathcal {O}_Y = \mathcal {O}_Y(-(n+k-1)E)$ . Then the canonical map

is zero in the derived category. Hence if R is a Noetherian reduced blowup-square splinter (e.g., Du Bois, lim-perfectoid pure, or Cohen-Macaulay and lim-perfectoid injective), then for every integer $k> 0$ ,

$$\begin{align*}\overline{J^{n+k}} \subseteq J^k. \end{align*}$$

The same holds if $J \subseteq R$ is an ideal that locally has analytic spread $\leq n$ .

Furthermore, if R is additionally normal and Nagata (e.g., excellent), one obtains that

$$\begin{align*}J_{>n+k-1} \subseteq J^k \end{align*}$$

where the left side is the fractional power integral closure as in [Reference Huneke and SwansonHS06, Definition 10.5.3].

Proof. Set $F = E_{\operatorname {\mathrm {{red}}}}$ with associated ideal sheaf $I_F$ , and note it may not be a divisor since Y need not be normal or even Noetherian. Since $ \mathcal {O}_Y(-(n+k-1)E) \to L^k(\underline {f}) \otimes \mathcal {O}_Y $ is zero in $D(Y)$ by Theorem 2.2, so is the map

$$ \begin{align*}{\mathbb R}\Gamma(Y,\mathcal{O}_Y(-(n+k)E)) \to L^k(\underline{f}) \otimes^{{\mathbb L}}_R {\mathbb R}\Gamma(Y,\mathcal{O}_Y(-E)). \end{align*} $$

We have a canonical map and now we see that the composition

is zero in the derived category, proving the first statement. For the second statement, use the splitting and take zeroth cohomology. The analytic spread statement follows as in the proof of Theorem 3.12 as blowups, and hence , commute with base change to the strict Henselization.

For the final statement in the normal Nagata case, replace Y by the blowup of $\overline {J^m}$ for $m \gg 0$ , which dominates the original Y and is also normal. Note that F is a reduced Weil divisor and we have

$$\begin{align*}\mathcal{O}_Y(-(n+k-1)E - F) \to L^k(\underline{f}) \otimes_R \mathcal{O}_Y(-F) \end{align*}$$

is zero in $D(Y)$ (note that while $\mathcal {O}_Y(-F)$ may not be locally free, every other term is). Arguing as above, we obtain that the composition

is zero in the derived category and hence we see that $\Gamma (Y, \mathcal {O}_Y(-(n+k-1)E - F)) \subseteq J^k$ . But the left side is $J_{>n+k-1}$ as R and Y are normal.

The following immediate corollary recovers and generalizes the multiplicity bound of rational, F-rational, F-pure and Du Bois singularities established in [Reference Huneke and WatanabeHW15, Reference ShibataShi17, Reference ParkPar25].

Corollary 3.19. Let $(R,\mathfrak {m})$ be a Noetherian reduced local ring of dimension d and embedding dimension e.

(a) If R is a birational derived splinter (e.g., pseudo-rational), then $e(R)\leq \binom {e-1}{d-1}$ .
(b) If R is a reduced blowup-square splinter (e.g., Du Bois, lim-perfectoid pure, or Cohen-Macaulay and lim-perfectoid injective), then $e(R)\leq \binom {e}{d}$ .

Proof. The proof is essentially the same as that of [Reference Huneke and WatanabeHW15, Theorem 3.1]. Without loss of generality, we may assume $R/\mathfrak {m}$ is infinite (see Remark 3.8). Let $J:=(x_1,\dots , x_d)$ be a minimal reduction of $\mathfrak {m}$ and we extend it to a minimal generating set $(x_1,\dots ,x_d,x_{d+1},\dots ,x_e)$ of $\mathfrak {m}$ . By Theorem 3.12 and Theorem 3.18, we know that $\mathfrak {m}^d\subseteq J$ in case (a) and $\mathfrak {m}^{d+1}\subseteq J$ in case (b). It follows that $R/J$ is generated by monomials in $x_{d+1},\dots ,x_e$ of degree $\leq d-1$ in case (a) and $\leq d$ in case (b). In particular, we have

$$\begin{align*}e(R)=e(J)\leq \ell(R/J)\leq \begin{cases} \binom{e-1}{d-1} \text{ in case (a),} \\ \binom{e}{d} \text{ in case (b)}. \end{cases} \end{align*}$$

3.3 Briançon-Skoda theorem for perfectoid rings

Since our Theorem 2.2 does not require R to be Noetherian, it has the following interesting consequence on integral closure of ideals in perfectoid rings. We refer the reader to [Reference Bhatt and ScholzeBS22] for unexplained terminology in what follows.

Corollary 3.20. Let R be a perfectoid ring and $J=(f_1,\dots ,f_n) \subseteq R$ . Then for every $k\geq 1$ , we have

$$ \begin{align*}J_{\operatorname{\mathrm{{perfd}}}}\overline{J^{n+k-1}} \subseteq J^k.\end{align*} $$

Furthermore, for every $k\geq 1$ , we have

$$ \begin{align*}\overline{J^{n+k}}\subseteq J^k.\end{align*} $$

Proof. Let $Y\to \mathrm {Spec} \,(R)$ be the blowup of $\overline {J^{n+k-1}}$ . By Theorem 2.2, $\overline {J^{n+k-1}}$ maps to zero in

$$\begin{align*}H_0\big(L^k(\underline{f})\otimes_R^{\mathbb{L}} {\mathbb R}\Gamma(Y,\mathcal{O}_Y)\big).\end{align*}$$

As R is perfectoid, $Y^{\text {pfd}}\to \mathrm {Spec} \,(R)^{\text {pfd}}=\mathrm {Spec} \,(R)$ is a J-almost isomorphism ([Reference Bhatt and ScholzeBS22, Corollary 8.12]). Hence $R\to {\mathbb R}\Gamma (Y,\mathcal {O}_Y)$ is J-almost split (since it factors the map to ${\mathbb R}\Gamma (Y^{\text {pfd}},\mathcal {O}_{Y, \text {perfd}})$ which is J-almost isomorphic to R). Therefore, $\overline {J^{n+k-1}}$ has J-almost zero image in $H_0(L^k(\underline {f}))=R/J^k$ as wanted.

For the second statement, note that since perfectoidization is an arc-sheaf, we know that R is a direct summand of $Mathematical fraction with D dot in the numerator all over J sub n plus k minus 1 in the denominator.$ (see [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMP+24, Lemma 5.3]). By Theorem 3.18 (whose first statement does not assume R is Noetherian),

is zero in $D(R)$ . Now use the splitting and take zeroth cohomology to obtain that $\overline {J^{n+k}}\subseteq J^k$ .

4 Connections with closure operations

Inspired by [Reference Epstein, McDonald, Rebecca and SchwedeEMRS25], we make the following definition.

Definition 4.1. Suppose that R is a reduced Noetherian ring and $J \subseteq R$ is an ideal. We define the birational preclosure of J, denoted $J^{\operatorname {\mathrm {Bir}}}$ , to be the union

$$\begin{align*}\bigcup_Y \ker\Big(R \to H_0\big(R/J \otimes^{{\mathbb L}} {\mathbb R}\Gamma(Y, \mathcal{O}_Y)\big) \Big) \end{align*}$$

where $Y \to \mathrm {Spec} \, R$ varies over all proper birational maps. As the set of proper birational morphisms is filtered, we see that this union is in fact an ideal.

Remark 4.2. For general non-Noetherian R, we expect that one should replace the birational condition of the birational preclosure with the M-morphisms of [Reference LyuLyu22].

Since R is Noetherian, $J^{\operatorname {\mathrm {Bir}}}$ is computed by a single Y or, by Chow’s lemma, a single blowup. Furthermore, the formation of $J^{\operatorname {\mathrm {Bir}}}$ commutes with localization since blowups can be spread out. If $R \to S$ is finite birational, we may take our Y factoring through $\mathrm {Spec} \, S \to \mathrm {Spec} \, R$ . Hence we obtain:

Lemma 4.3. Suppose R is a reduced Noetherian ring, and $R \hookrightarrow S$ is a finite extension such that the induced $\mathrm {Spec} \, S \to \mathrm {Spec} \, R$ is birational (e.g., if $S = \prod _i R/Q_i$ where the $Q_i$ are the minimal primes of R). Then $ J^{\operatorname {\mathrm {Bir}}} = (JS)^{\operatorname {\mathrm {Bir}}} \cap R. $

We note the following immediate consequence of the definition of birational derived splinters.

Lemma 4.4. If R is a birational derived splinter, then $J^{\operatorname {\mathrm {Bir}}} = J$ for every ideal $J \subseteq R$ .

Given a resolution of singularities, $J^{\operatorname {\mathrm {Bir}}}$ is computed by that resolution.

Lemma 4.5. Suppose $\pi : X \to \mathrm {Spec} \, R$ is a proper birational map with X a global birational derived splinter. If $J \subseteq R$ is an ideal, then

$$\begin{align*}J^{\operatorname{\mathrm{Bir}}} = \ker\Big(R \to H_0(R/J \otimes^{{\mathbb L}} {\mathbb R}\Gamma(X, \mathcal{O}_X)) \Big). \end{align*}$$

Proof. For the main statement, it suffices to consider proper birational maps $Y \xrightarrow {\rho } X \xrightarrow {\pi } \mathrm {Spec} \, R$ , but $\mathcal {O}_X \to {\mathbb R}\rho _* \mathcal {O}_Y$ splits.

Remark 4.6. We saw that pseudo-rational schemes with a canonical sheaf are birational derived splinters by Lemma 3.9, and hence such X can be used in Lemma 4.5. Furthermore, if $X \to \mathrm {Spec} \, R$ is a resolution of singularities with X of finite dimension, one can use that X in Lemma 4.5. In fact, the finite dimensional hypothesis is not needed to prove that regular X is a derived splinter via an argument of Bhatt ([Reference Ma and SchwedeMS20, Remark 3.4], [Reference Lank and VenkateshLV24, Lemma 3.16]); we shall repeat Bhatt’s argument within the proof of Theorem 5.3 below. Recall resolutions of singularities are known to exist if R is excellent and either $\dim R \leq 3$ [Reference AbhyankarAbh66, Reference LipmanLip78, Reference CutkoskyCut09, Reference Cossart and PiltantCP08, Reference Cossart and PiltantCP09, Reference Cossart and PiltantCP19] or R has characteristic zero [Reference HironakaHir64, Reference TemkinTem08].

It is not clear to us whether or not $J^{\operatorname {\mathrm {Bir}}}$ is a closure operation. In other words:

Question 4.7. With notation as above, is $(J^{\operatorname {\mathrm {Bir}}})^{\operatorname {\mathrm {Bir}}} = J^{\operatorname {\mathrm {Bir}}}$ ?

However, we obtain the following result.

Proposition 4.8. Suppose R is a reduced Noetherian ring and $J = (f_1, \dots , f_n)$ is an ideal. Then $\overline {J^{n+k-1}} \subseteq (J^k)^{\operatorname {\mathrm {Bir}}}$ for any $k \geq 1$ . The same holds if $J \subseteq R$ is an ideal that locally has analytic spread $\leq n$ .

Proof. As both integral closure and birational closure can be computed modulo minimal primes, we may assume that R is a domain. If $J = 0$ , then the statement is clear. Set $Y \to \mathrm {Spec} \, R$ to be the blowup of $\overline {J^{n+k-1}}$ . This map is proper and birational since R is a domain and J is nonzero. We see there is a map from $L^k(\underline {f})$ to its bottom homology, $L^k(\underline {f}) \to R/J^k$ . Hence we see

$$\begin{align*}\begin{array}{rcl} \overline{J^{n+k-1}} & \subseteq & \ker\Big(R \to H_0\big( L^k(\underline{f}) \otimes^{{\mathbb L}} {\mathbb R}\Gamma(Y, \mathcal{O}_Y) \big) \Big)\\ & \subseteq & \ker\Big(R \to H_0\big( R/J^k \otimes^{{\mathbb L}} {\mathbb R}\Gamma(Y, \mathcal{O}_Y) \big) \Big)\\ & \subseteq & (J^k)^{\operatorname{\mathrm{Bir}}}. \end{array} \end{align*}$$

StartLayout 1st Row StartLayout 1st Row 1st Column upper J Superscript n plus k minus 1 Baseline overbar 2nd Column subset of or equal to 3rd Column kernel left parenthesis upper R right arrow upper H 0 left parenthesis upper L Superscript k Baseline left parenthesis f underbar right parenthesis circled times Superscript double struck upper L Baseline double struck upper R normal upper Gamma left parenthesis upper Y comma script upper O Subscript upper Y Baseline right parenthesis right parenthesis right parenthesis 2nd Row 1st Column Blank 2nd Column subset of or equal to 3rd Column kernel left parenthesis upper R right arrow upper H 0 left parenthesis upper R divided by upper J Superscript k Baseline circled times Superscript double struck upper L Baseline double struck upper R normal upper Gamma left parenthesis upper Y comma script upper O Subscript upper Y Baseline right parenthesis right parenthesis right parenthesis 3rd Row 1st Column Blank 2nd Column subset of or equal to 3rd Column left parenthesis upper J Superscript k Baseline right parenthesis Superscript upper B i r Baseline period EndLayout EndLayout

For the second statement, as the statement is local (indeed, the formation of $J^{\operatorname {\mathrm {Bir}}}$ commutes with localization), we assume that R is local. Then one argues again as in Theorem 3.12 to replace J by its minimal reduction.

Proposition 4.8 implies many other well-known closure-versions of the Briançon-Skoda theorem.

4.1 Characteristic zero

In combination with Lemma 4.5, Proposition 4.8 gives the following answer to [Reference Epstein, McDonald, Rebecca and SchwedeEMRS25, Question 6.7].

Corollary 4.9. Let R be an excellent ring of characteristic zero and $J=(f_1,\dots ,f_n)$ . Then

$$\begin{align*}(J^k)^{\operatorname{\mathrm{Bir}}} = (J^k)^{\operatorname{\mathrm{Hir}}} := \ker\big(R \to H_0(R/J^k \otimes^{{\mathbb L}} {\mathbb R}\Gamma(X, \mathcal{O}_X))\big) \end{align*}$$

where $X \to \mathrm {Spec} \, R$ is any resolution of singularities. Hence $\overline {J^{n+k-1}}\subseteq (J^k)^{\operatorname {\mathrm {Hir}}} $ for all $k\geq 1$ .

Statements for other characteristic zero closures follow from the characteristic $p> 0$ results below.

4.2 Characteristic $p> 0$

Suppose R is a reduced Noetherian ring of characteristic $p> 0$ . Let $R^+ = \prod _Q (R/Q)^+$ where Q runs over the minimal primes of R. For any ideal $J \subseteq R$ , we denote by $J^+ := (J R^+) \cap R$ . Note $J^+ \subseteq J^*$ by [Reference Hochster and HunekeHH90, Lemma 4.11].

Proposition 4.10 [Reference Hochster and HunekeHH95, Theorem 7.1]

Suppose R is a reduced Noetherian ring and $I \subseteq R$ is any ideal. Then $I^{\operatorname {\mathrm {Bir}}} \subseteq I^+$ . As a consequence, if $J = (f_1, \dots , f_n) \subseteq R$ , then we have

$$\begin{align*}\overline{J^{{n+k-1}}} \subseteq (J^k)^+ \end{align*}$$

for every integer $k> 0$ . Hence we also obtain $\overline {J^{{n+k-1}}} \subseteq (J^k)^*$ recovering [Reference Hochster and HunekeHH90, Theorem 5.4].

Proof. As R is Noetherian, we may fix $\pi : Y \to X$ proper birational computing $J^{\operatorname {\mathrm {Bir}}}$ . By [Reference BhattBha12, Theorem 0.4], there is a map ${\mathbb R}\Gamma (Y, \mathcal {O}_Y) \to R^+$ factoring the inclusion $R \to R^+$ . Hence we immediately see that

$$\begin{align*}\ker\Big(R \to H_0(R/J^k \otimes^{{\mathbb L}} {\mathbb R}\Gamma(Y, \mathcal{O}_Y)) \Big) \subseteq \ker\Big(R \to R/J^k \otimes R^+ \Big) = (J^k)^+. \end{align*}$$

The second and third statements follow immediately.

4.3 Mixed characteristic

The same argument goes through without substantial change in the mixed characteristic case thanks to [Reference BhattBha20]. In fact, Proposition 4.11 below strictly contains Proposition 4.10 since the p-adic completion is a trivial operation in characteristic p.

Let R denote a Noetherian reduced ring with p in its Jacobson radical. By $\widehat {R^+}$ , we mean the p-adic completion of $R^+ = \prod _Q (R/Q)^+$ where again Q runs over the minimal primes of R.

Proposition 4.11 [Reference Rodríguez-Villalobos and SchwedeRS24, Reference HeitmannHei01]

Suppose R is a reduced Noetherian ring with p in its Jacobson radical and suppose that $I \subseteq R$ is an ideal. Then $I^{\operatorname {\mathrm {Bir}}} \subseteq I\widehat {R^+} \cap R \subseteq I^{\mathrm {epf}}$ where $\widehat {R^+}$ denotes the p-adic completion of $R^+$ and $\operatorname {\mathrm {epf}}$ denotes the full extended plus closure.

As a consequence, if $J = (f_1, \dots , f_n) \subseteq R$ , then

$$\begin{align*}\overline{J^{{n+k-1}}} \subseteq (J^k) \widehat{R^+} \cap R \end{align*}$$

recovering [Reference Rodríguez-Villalobos and SchwedeRS24, Theorem 4.1]. Hence in an arbitrary ring, we also recover $\overline {J^{n+k-1}} \subseteq (J^k)^{\operatorname {\mathrm {ep}}}$ of [Reference HeitmannHei01, Theorem 4.2] where $\operatorname {\mathrm {ep}}$ denotes the extended plus closure.

Proof. We may assume that R is a domain as all objects can be computed modulo minimal primes. We then assume that R has mixed characteristic $(0, p>0)$ as the equal characteristic $p> 0$ case was done above. Suppose $Y \to \mathrm {Spec} \, R$ is proper birational. By [Reference BhattBha20, Theorem 3.12] and derived Nakayama, the derived completion of ${\mathbb R}\Gamma (Y^+, \mathcal {O}_{Y^+})$ coincides with $\widehat {R^+}$ . Hence we have a factorization

$$\begin{align*}R \to {\mathbb R}\Gamma(Y, \mathcal{O}_Y) \to \widehat{R^+}. \end{align*}$$

The first result follows. Note $I \widehat {R^+} \cap R \subseteq I^{\operatorname {\mathrm {epf}}}$ essentially by definition of full extended plus closure in [Reference HeitmannHei01]. For the statement about extended plus closure $(-)^{\operatorname {\mathrm {ep}}}$ , by using [Reference HeitmannHei01, Proposition 1.2 and Lemma 2.3] we may reduce to the case of a local domain essentially of finite type over $\mathbb {Z}$ with some prime $p> 0$ in its maximal ideal. In particular, $\operatorname {\mathrm {ep}}$ coincides with $\operatorname {\mathrm {epf}}$ in this case.

5 Uniform Briançon-Skoda and uniform Artin-Rees

In this section, we prove two conjectures of Huneke [Reference HunekeHun92, Conjecture 1.3 and Conjecture 1.4] as a major application of our results and methods in previous sections. We first give an outline of the strategy. Suppose R is a domain and $\pi : X \to \mathrm {Spec} \, R$ is a proper surjective map with X an integral global birational derived splinter (e.g., $\pi $ could be a regular alteration). Suppose we have a map $\psi : {\mathbb R}\Gamma (X, \mathcal {O}_X) \to R$ such that the composition

$$ \begin{align*}R \to {\mathbb R}\Gamma(X, \mathcal{O}_X) \xrightarrow{\psi} R\end{align*} $$

is multiplication by $c \neq 0$ (such maps always exist since ${\mathbb R}\Gamma (X, \mathcal {O}_X) \in D^b_{\operatorname {\mathrm {coh}}}(R)$ and since such maps exist generically). Let $I = (f_1, \dots , f_n)\subseteq R$ and set $Y' \to X$ to be the blowup of $\overline {I^{n+k-1}} \mathcal {O}_X$ (so that the composition $Y' \to \mathrm {Spec} \, R$ factors through $Y \to \mathrm {Spec} \, R$ , the blowup of $\overline {I^{n+k-1}}$ ), then the argument from Remark Remark 3.13 immediately implies that we have $ c \overline {I^{n+k-1}} \subseteq I^k. $ Of particular importance here is that c does not depend on I and k. The existence of such a uniform c, which forces $\mathrm {T}_n(R) := \bigcap _{m,I} (I^{m - n} : \overline {I^m})$ to be nonzero for some n (which could be taken as $\dim R$ eventually, by our method), is one of Huneke’s key tools for proving uniform Briançon-Skoda and uniform Artin-Rees properties in [Reference HunekeHun92].

It was known, for rings that are quotients of regular rings, that the existence of projective resolutions of singularities implies these uniformity statements via a different argument, see [Reference HunekeHun00]. Furthermore, in unpublished work, Datta showed that Huneke’s argument also goes through assuming only that projective regular alterations exist [Reference DattaDat25].

However, the existence of regular alterations is not known in full generality for excellent domains. Instead, we adapt the argument above to use Gabber’s weak local uniformization ([Reference Illusie, Laszlo and OrgogozoILO14, Exposé VII, Theorem 1.1]), and instead of using regular alterations, we use a certain diagram of schemes. The other key ingredient in Huneke’s recipe, that $0 \neq \mathrm {CM}(R)$ , where $\mathrm {CM}(R)$ is the set of elements of R that annihilate the homology of complexes of finitely generated free modules satisfying the standard conditions on height and rank, was proven in [Reference ZhouZho07]. Alternatively, this also follows quickly by combining [Reference LyuLyu25] and [Reference KawasakiKaw00] up to a faithfully flat extension which is enough for us.

We begin with a lemma about truncated hypercovers in the alteration topology.

Lemma 5.1. Let X be an irreducible, quasi-excellent Noetherian scheme. Then for every integer $n\geq 0$ , we can construct a commutative diagram

Commutative diagram: n V sub less than or equal to n arrows to n V sub dot arrows to n V. Below, X sub less than or equal to n arrows to X sub dot arrows to X. Vertical arrows point from the top row to the bottom row at the left and right ends.

where ${^n}V\to X$ is an alteration, ${^n}V_{\bullet }\to {^n}V$ is a Zariski hypercover and $X_{\bullet }\to X$ is a hypercover in the alteration topology with each $X_i$ regular. Here $V_{\leq n}$ and $X_{\leq n}$ are the n-truncations of ${^n}V_{\bullet }$ and $X_{\bullet }$ respectively.

Proof. We will repeatedly use Gabber’s weak local uniformization theorem [Reference Illusie, Laszlo and OrgogozoILO14, Exposé VII, Theorem 1.1]. We begin by selecting a cover $\{X_0^i\to X\}$ in the alteration topology so that each $X_0^i$ is regular by weak local uniformization. By [Reference Illusie, Laszlo and OrgogozoILO14, Exposé II, Theorem 3.2.1], there exists an alteration ${^0}V\to X$ and a Zariski open cover ${^0}V^i$ of ${^0}V$ which fits into the following commutative diagram

Commutative diagram where super 0 V super i maps right to super 0 V and down to X sub 0 super i. From super 0 V, an arrow points down to X. From X sub 0 super i, an arrow points right to X.

Next, we find a cover $\{{^0}Y^j\to {^0}V\}$ in the alteration topology so that each ${^0}Y^j$ is regular by weak local uniformization. Again by [Reference Illusie, Laszlo and OrgogozoILO14, Exposé II, Theorem 3.2.1], there exists an alteration ${^1}V\to {^0}V$ and a Zariski open cover ${^1}\widetilde {V}^j$ of ${^1}V$ which fits into the following commutative diagram

Commutative diagram. Top-left 1 V-tilde super j arrows right to top-right 1 V and down to bottom-left 0 Y super j. Top-right 1 V arrows down to bottom-right 0 V. Bottom-left 0 Y super j arrows right to bottom-right 0 V.

At this point, we note that each ${{}^{0}V^{a}} \times _{{{}^0}V} {{{}^0}V{^b}}$ admits a regular cover in the alteration topology

$$ \begin{align*}\{{{}^0}Y^j\times_{{{}^0}V}({{}^0}V^{a} \times_{{{}^0}V} {{}^0}V^{b}) \to {{}^0}V^{a} \times_{{{}^0}V} {{}^0}V^{b}\}.\end{align*} $$

Now, we set ${}{{}^1}V^{i}:= {{}^0}V^{i}\times _{{{}^0}V}^{1}V$ (a Zariski open subset of ${{}^1}V$ ). By the commutative diagram above and base change, we find that

$$ \begin{align*}\{{{}^1}\widetilde{V}^j \times_{{{}^1}V}({{}^1}V^{a} \times_{{{}^1}V} {{}^1}V^{b}) \to {{}^1}V^{a} \times_{{{}^1}V} {{}^1}V^{b}\}\end{align*} $$

is a Zariski open cover and that each ${{}^1}\widetilde {V}^j \times _{{{}^1}V}({{}^1}V^{a} \times _{{{}^1}V} {{}^1}V^{b})$ factors through the regular scheme ${{}^0}Y^j\times _{{{}^0}V}({{}^0}V^{a} \times _{{{}^0}V} {{}^0}V^{b})$ , which in turn maps to $X_0^a \times ^{\text {alt}}_X X_0^b$ (which is the closed subscheme of $X_0^a\times _XX_0^b$ made up of components that dominate X; in particular, it maps to $X_0^a\times _XX_0^{b}$ ).Footnote ⁶ In sum, we find that ${{}^1}V$ has a $1$ -truncated Zariski hypercover ${{}^1}V_{\leq 1} \to {{}^1}V$ with

$$ \begin{align*}{{}^1}V_0=\{{{}^1}V^{i}\}, \text{ and } {{}^1}V_1=\{{{}^1}\widetilde{V}^j \times_{{{}^1}V}({{}^1}V^{a} \times_{{{}^1}V} {{}^1}V^{b})\}.\end{align*} $$

Moreover, this ${{}^1}V_{\leq 1}$ dominates the following $1$ -truncated alteration hypercover $X_{\leq 1}$ by regular schemes, where

$$ \begin{align*}X_0=\{X^i_0\}, \text{ and } X_1=\{{{}^0}Y^j\times_{{{}^0}V}({{}^0}V^{a} \times_{{{}^0}V} {{}^0}V^b) \}.\end{align*} $$

Note that as written, the above is not technically a hypercover, as it only has face maps and no degeneracy maps, and hypercovers are simplicial objects. However, using the notion of a split simplicial object (see for instance [Reference DeligneDel71, 6.2.2], [SGA72, ex Vbis 5.1] and [Reference LeeLee09, Remark 2.14]), we only need to specify the nondegenerate simplices appearing in any degree in order to define a unique simplicial object. More precisely, in the language of [Reference ConradCon03, Theorem 4.12], given a split n-truncated simplicial scheme W and a collection N that we would like to adjoin as nondegenerate $(n+1)$ -simplices and a map $N\to (\text {cosk}_nW)_{n+1}$ , there is a unique, up to isomorphism, split $(n+1)$ -truncated simplicial scheme $W'$ such that $W^{\prime }_{\leq n}=W_{\leq n}$ and the nondegenerate $(n+1)$ simplices of $W'$ are N. Thus, we will abuse notation and specify only the nondegenerate simplices in each degree in what follows.

We will now essentially repeat this procedure to inductively construct ${^m}V$ , $m\geq 1$ , with an m-truncated Zariski hypercover ${{}^m}V_{\leq m} \to {{}^m}V$ that dominates an m-truncated alteration hypercover $X_{\leq m}$ so that each $X_i$ is regular. Indeed, suppose ${{}^m}V$ is constructed, we use weak local uniformization to obtain a regular cover $\{{{}^m}Y^j\to {{}^m}V\}$ in the alteration topology, and find ${{}^{m+1}}V$ an alteration over ${{}^m}V$ together with a Zariski open cover $\{{{}^{m+1}}\widetilde {V}^j\}$ dominating $\{{{}^m}Y^j\}$ . The m-truncated Zariski hypercover ${{}^m}V_{\leq m}$ pulls back to an m-truncated Zariski hypercover ${{}^{m+1}}V_{\leq m}$ of ${{}^{m+1}}V$ . For each element $U \in (\text {cosk}_{m}({{}^{m+1}}V_{\leq m}))_{m+1}$ , noting it is the pullback of some $U' \in (\text {cosk}_{m}({{}^{m}}V_{\leq m}))_{m+1}$ , consider the open cover

$$ \begin{align*}\{{{}^{m+1}}\widetilde{V}^j\times_{{{}^{m+1}}V}U\to U\}.\end{align*} $$

The collection of all these $\{{{}^{m+1}}\widetilde {V}^j\times _{{{}^{m+1}}V}U\}_{U,j}$ thus form a Zariski open cover of $(\text {cosk}_{m}({{}^{m+1}}V_{\leq m}))_{m+1}$ . Moreover, each ${{}^{m+1}}\widetilde {V}^j\times _{{{}^{m+1}}V}U$ maps to the regular scheme ${{}^m}Y^j \times _{{{}^m}V}U'$ . In sum, we have

Commutative diagram: {m+1 V~j x U} maps to (cosk m V<=m)m+1 and down to {m Yj x U’}. {m Yj x U’} maps to (cosk m V<=m)m+1 and diagonally to (cosk m X<=m)m+1.

Here $(\text {cosk}_{m}({}^{m}X_{\leq m}))_{m+1}$ is computed in the alteration topology (so it is built by fiber products in the alteration topology), and the dotted arrow above is a cover in the alteration topology. We now set ${{}^{m+1}}V_{m+1} := \{{{}^{m+1}}\widetilde {V}^j\times _{{{}^{m+1}}V}U\}_{U,j}$ and $X_{m+1}:= \{{{}^m}Y^j \times _{{{}^m}V}U'\}_{U', j}$ . It is then readily checked that we have the following commutative diagram

Commutative diagram where m+1 V sub less than or equal to m+1 maps right to m+1 V and down to X sub less than or equal to m+1. Both m+1 V and X sub less than or equal to m+1 map to X.

Finally, we set ${{}^n}V_{\bullet } := \text {cosk}_{n}({}^nV_{\leq n})$ and we extend $X_{\leq n}$ to an alteration hypercover $X_{\bullet }$ so that each $X_i$ is regular by weak local uniformization. Note that ${}^nV_{\bullet }$ is a Zariski hypercover of ${}^nV$ . We obtain the desired commutative diagram

Commutative diagram. Top row: n V sub less than or equal to n arrows to n V sub bullet arrows to n V. Bottom row: X sub less than or equal to n arrows to X sub bullet arrows to X. Vertical arrows connect top terms to bottom terms.

Note that there might not be a map from ${}^nV_{\bullet }\to X_{\bullet }$ fitting the commutative diagram above though, since we built ${}^nV_{\bullet }$ out of $V_{\leq n}$ while one might need further simplices to build $X_{\bullet }$ to keep the terms nonsingular.

Remark 5.2. In the proof of Lemma 5.1, instead of writing a Zariski open cover $\{{{}^{m+1}}\widetilde {V}^j\}$ in each step, we could use a single cover ${{}^{m+1}}\widetilde {V}:= \coprod _j{{}^{m+1}}\widetilde {V}^j$ to reduce the excess of indices and simplify the notation (the proof will not be altered too much). We decided to write down the open cover and keep the additional index j to make the construction clear and to remind ourselves that each ${{}^{m+1}}\widetilde {V}^j$ dominates a ${{}^{m}}{Y}^j$ (which is based on Gabber’s weak local uniformization).

We are now ready to prove our main result which will be the key ingredient in proving Huneke’s conjecture.

Theorem 5.3. Let R be an excellent domain of dimension d. Then ${\mathrm T}_{d}(R) \neq 0$ .

Proof. First, note that we can replace R by $R\otimes _{\mathbb {Z}}\mathbb {Z}(t)$ (where $\mathbb {Z}(t)$ denotes $\mathbb {Z}[t]$ localized at the multiplicative set of all elements not contained in any $p \mathbb {Z}[t]$ ) to assume that all residue fields of R are infinite.

Showing that $\text {T}_{d}(R)=\bigcap _{I, n}(I^{n-d}: \overline {I^n})$ is nonzero is the same as showing that $\bigcap _{I, k\geq 1}(I^{k}: \overline {I^{d+k}})$ is nonzero. Since R has infinite residue fields, every ideal $I\subseteq R$ admits a minimal reduction J generated by at most $d+1$ elements by [Reference Huneke and SwansonHS06, Theorem 8.7.3]. By Theorem 2.2, if Y is the normalized blowup of J (which agrees with the blowup of $\overline {J^N}$ for some $N\gg 0$ ), then the natural map

$$ \begin{align*}\overline{I^{d+k}} \to R\to {\mathbb R}\Gamma(Y, \mathcal{O}_Y) \to L^k(J)\otimes^{{\mathbb L}} {\mathbb R}\Gamma(Y, \mathcal{O}_Y)\end{align*} $$

is the zero map in $D(R)$ .

Let $X=\mathrm {Spec} \, R$ . Applying Lemma 5.1 with $n=d+2$ and setting $V:={}^{d+2}V$ and $V_{\bullet }:= {}^{d+2}V_{\bullet }$ , we have a commutative diagram

Commutative diagram. Top row: V sub less than or equal to d+2 arrows to V bullet arrows to V. Bottom row: X sub less than or equal to d+2 arrows to X bullet arrows to X. Vertical arrows connect the first and last terms of each row.

where $V_{\bullet }$ is a Zariski hypercover of V, $V\to X$ is an alteration, and $X_{\bullet }\to X$ is an alteration hypercover of X with each $X_i$ regular.

Let $Y_i$ be the blowup of $\overline {J^N}\mathcal {O}_{X_i}$ for each $X_i$ . We have a commutative diagram of diagrams of schemes:

Commutative diagram. Top row Y sub dot arrow to Y. Bottom row X sub dot arrow to X. Vertical arrows point from Y sub dot to X sub dot on the left and from Y to X on the right.

which induces a commutative diagram:

$Commutative diagram. Top: I^{d+k} → R → RΓ(Y, O_Y) → L^k(J) ⊗ RΓ(Y, O_Y). Bottom: RΓ(X., O_X.) → RΓ(Y., O_Y.) → L^k(J) ⊗ RΓ(Y., O_Y.). Vertical arrows map R to RΓ(X., O_X.), and RΓ(Y, O_Y) terms to their simplicial counterparts.$

In particular, the diagram guarantees that the natural map

$$ \begin{align*}\overline{I^{d+k}} \to R\to {\mathbb R}\Gamma(X_{\bullet}, \mathcal{O}_{X_{\bullet}}) \to L^k(J)\otimes^{{\mathbb L}} {\mathbb R}\Gamma(Y_{\bullet}, \mathcal{O}_{Y_{\bullet}})\end{align*} $$

is zero in $D(R)$ .

We now repeat an argument of Bhatt’s ([Reference Ma and SchwedeMS20, Remark 3.4], [Reference Lank and VenkateshLV24, Lemma 3.16]). We note that since each $X_i$ is regular, each ${\mathbb R}{\pi _i}_*\mathcal {O}_{Y_i} \in D_{\text {perf}}(X_i)$ and is a commutative algebra object in $D(X_i)$ (here $\pi _i$ denotes the map $Y_i\to X_i$ ). Therefore, we have canonical maps

$$ \begin{align*}\mathcal{O}_{X_i} \to {\mathbb R}{\pi_i}_*\mathcal{O}_{Y_i} \xrightarrow{s} {\mathbb R}{\mathcal{H}\mathrm{om}}_{\mathcal{O}_{X_i}}({\mathbb R}{\pi_i}_*\mathcal{O}_{Y_i}, {\mathbb R}{\pi_i}_*\mathcal{O}_{Y_i}) \xrightarrow{t} \mathcal{O}_{X_i}\end{align*} $$

where s comes from the commutative algebra structure on ${\mathbb R}{\pi _i}_*\mathcal {O}_{Y_i}$ and t is the trace map coming from the perfectness of ${\mathbb R}{\pi _i}_*\mathcal {O}_{Y_i}$ . Moreover, the composition map is generically the identity and hence the identity map (note that by construction, all the (irreducible components of) $X_i$ ’s and $Y_i$ ’s are generically finite over and dominate $X = \mathrm {Spec} \, R$ ). As s and t are functorial, these induce maps

$$ \begin{align*}\mathcal{O}_{X_{\bullet}} \to {\mathbb R}{\pi_{\bullet}}_*\mathcal{O}_{Y_{\bullet}} \to \mathcal{O}_{X_{\bullet}}\end{align*} $$

such that the composition is the identity. It follows that the canonical map ${\mathbb R}\Gamma (X_{\bullet }, \mathcal {O}_{X_{\bullet }}) \to {\mathbb R}\Gamma (Y_{\bullet }, \mathcal {O}_{Y_{\bullet }})$ splits in $D(R)$ . As a consequence, the natural map

$$ \begin{align*}\overline{I^{d+k}} \to R\to {\mathbb R}\Gamma(X_{\bullet}, \mathcal{O}_{X_{\bullet}}) \to L^k(J)\otimes^{{\mathbb L}} {\mathbb R}\Gamma(X_{\bullet}, \mathcal{O}_{X_{\bullet}})\end{align*} $$

is zero in $D(R)$ . We thus see that the composition

$$ \begin{align*} \overline{I^{d+k}} \to & R\to {\mathbb R}\Gamma(X_{\bullet}, \mathcal{O}_{X_{\bullet}}) \to L^k(J)\otimes^{{\mathbb L}} {\mathbb R}\Gamma(X_{\bullet}, \mathcal{O}_{X_{\bullet}}) \\ & \to L^k(J)\otimes^{{\mathbb L}} {\mathbb R}\Gamma(X_{\leq d+2}, \mathcal{O}_{X_{\leq d+2}})\to L^k(J)\otimes^{{\mathbb L}} {\mathbb R}\Gamma(V_{\leq d+2}, \mathcal{O}_{V_{\leq d+2}}) \end{align*} $$

is zero in $D(R)$ . Since J is generated by at most $d+1$ elements, the Buchsbaum-Eisenbud complex $L^k(J)$ has length at most $d+1$ . It follows that (since $V_{\bullet }\to V$ is a Zariski hypercover)

$$\begin{align*}\begin{array}{rl} & H_0\big(L^k(J) \otimes^{{\mathbb L}} {\mathbb R}\Gamma(V,\mathcal{O}_{V})\big)\\ \cong & H_0\big(L^k(J) \otimes^{{\mathbb L}} {\mathbb R}\Gamma(V_{\bullet},\mathcal{O}_{V_{\bullet}})\big)\\ \cong & H_0\big(L^k(J) \otimes^{{\mathbb L}} {\mathbb R}\Gamma(V_{\leq d+2},\mathcal{O}_{V_{\leq d+2}})\big) \end{array} \end{align*}$$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column upper H 0 left parenthesis upper L Superscript k Baseline left parenthesis upper J right parenthesis circled times Superscript double struck upper L Baseline double struck upper R normal upper Gamma left parenthesis upper V comma script upper O Subscript upper V Baseline right parenthesis right parenthesis 2nd Row 1st Column approximately equals 2nd Column upper H 0 left parenthesis upper L Superscript k Baseline left parenthesis upper J right parenthesis circled times Superscript double struck upper L Baseline double struck upper R normal upper Gamma left parenthesis upper V Subscript bullet Baseline comma script upper O Subscript upper V Sub Subscript bullet Subscript Baseline right parenthesis right parenthesis 3rd Row 1st Column approximately equals 2nd Column upper H 0 left parenthesis upper L Superscript k Baseline left parenthesis upper J right parenthesis circled times Superscript double struck upper L Baseline double struck upper R normal upper Gamma left parenthesis upper V Subscript less than or equals d plus 2 Baseline comma script upper O Subscript upper V Sub Subscript less than or equals d plus 2 Subscript Baseline right parenthesis right parenthesis EndLayout EndLayout

where the second isomorphism follows from a simple spectral sequence argument. Thus after taking $H_0(-)$ , we find that the map

$$ \begin{align*}\overline{I^{d+k}} \to H_0\big(L^k(J) \otimes^{{\mathbb L}} {\mathbb R}\Gamma(V_{\leq d+2},\mathcal{O}_{V_{\leq d+2}})\big)\cong H_0\big(L^k(J) \otimes^{{\mathbb L}} {\mathbb R}\Gamma(V,\mathcal{O}_{V})\big)\end{align*} $$

is zero. However, since V is proper surjective over X, we can pick $c\neq 0$ so that multiplication by c on R factors through ${\mathbb R}\Gamma (V,\mathcal {O}_{V})$ (since $R\to {\mathbb R}\Gamma (V,\mathcal {O}_{V})$ splits after tensoring with the fraction field of R and ${\mathbb R}\Gamma (V,\mathcal {O}_{V})\in D^b_{\text {coh}}(R)$ ). Note that c depends only on V and does not depend on I or k. We thus find that $c\overline {I^{d+k}}\subseteq J^k\subseteq I^k$ for all I and all $k\geq 1$ , that is, $\text {T}_{d}(R)\ni c\neq 0$ as desired.

Corollary 5.4 (Uniform Briançon-Skoda)

Suppose R is a quasi-excellent reduced Noetherian ring of finite dimension. Then there exists a positive integer k such that for all ideals $I\subseteq R$ , $\overline {I^n}\subseteq I^{n-k}$ for all integers $n\geq k$ .

Proof. First of all, we can replace R by $R\otimes _{\mathbb {Z}}\mathbb {Z}(t)$ (where $\mathbb {Z}(t)$ denotes $\mathbb {Z}[t]$ localized at the multiplicative set of all elements not contained in any $p \mathbb {Z}[t]$ ) to assume that all residue fields of R are infinite. Secondly, by replacing R by a faithfully flat étale cover, we may assume that R is excellent by [Reference LyuLyu25, Corollary 6.6] (Lyu pointed out to us this generalization from excellent to quasi-excellent).

By [Reference HunekeHun92, Proposition 3.7], it is enough to show that $\text {CM}(R/P)\neq 0$ and that $\text {T}(R/P) := \bigcup _k \mathrm {T}_k(R/P) \neq 0$ for all prime ideals $P\in \mathrm {Spec} \,(R)$ . We have just shown in Theorem 5.3 that $\text {T}(R/P)\neq 0$ . We know $\text {CM}(R/P) \neq 0$ by [Reference ZhouZho07]. Alternatively, by replacing R by a faithfully flat étale cover (which we already may have done when reducing to the excellent case), we may assume that R admits a dualizing complex by [Reference LyuLyu25, Theorem 6.5], and thus $R/P$ is a homomorphic image of a Gorenstein ring of finite dimension by [Reference KawasakiKaw00, Theorem 1.2] and so $\text {CM}(R/P)\neq 0$ by [Reference HunekeHun92, Proposition 4.5 (i)].

Remark 5.5. For any Noetherian local ring $(R,\mathfrak {m})$ of dimension d and any system of parameters $x_1,\dots ,x_d$ , it is easy to see that

$$ \begin{align*}\overline{(x_1^d,\dots,x_d^d)^{d-1}}\nsubseteq (x_1^d,\dots,x_d^d).\end{align*} $$

In particular, one cannot expect the conclusion of Corollary 5.4 to hold for infinite dimensional excellent rings (such examples exist, see [Reference TanakaTan19]): for if such k exists, then localizing at a prime of height $k+1$ would contradict the noncontainment above.

Corollary 5.6 (Uniform Artin-Rees)

$$\begin{align*}I^n M \cap N \subseteq I^{n-\ell} M. \end{align*}$$

Proof. By base change, we can again replace R by $R\otimes _{\mathbb {Z}}\mathbb {Z}(t)$ and so assume R has infinite residue fields and, as above, assume that R is excellent by [Reference LyuLyu25, Corollary 6.6]. For all $P\in \mathrm {Spec} \, R$ , we know that ${\mathrm {CM}}(R/P) \neq 0$ by [Reference ZhouZho07] (or alternatively as above). By Theorem 5.3, we see that $\mathrm {T}(R/P) \neq 0$ . Hence the result follows from [Reference HunekeHun92, Theorem 3.4].

Acknowledgments

This paper grew out of studying the Hironaka closure in [Reference Epstein, McDonald, Rebecca and SchwedeEMRS25] where Neil Epstein is also an author. We deeply thank Neil Epstein for many valuable conversations on the topic of this paper. The authors would also like to thank Rahul Ajit, Bhargav Bhatt, Rankeya Datta, Daniel Erman, Federico Galetto, Sean Howe, Craig Huneke, Annette Huber-Klawitter, Srikanth Iyengar, Sándor Kovács, Robert Lazarsfeld, Zhenqian Li, Shiji Lyu, Wenbo Niu, Thomas Polstra, Claudiu Raicu, Ilya Smirnov, Shunsuke Takagi and Keller VandeBogert for valuable discussions related to this project and comments on earlier drafts. We particularly thank Keller VandeBogert for explaining the Buchsbaum-Eisenbud complex to us, and especially thank Bhargav Bhatt for several conversations related to hypercovers. Finally, we thank the referee for numerous useful comments and suggestions.

Competing interests

The authors have no competing interests to declare.

Funding statement

The first author received support from NSF grants DMS-2302430 and DMS-2424441. The second author would like to acknowledge the support of the Pacific Institute for the Mathematical Sciences and a grant from the Simons Foundation International [SFI-MPS-T-Institutes-00020822, OY]. The third author received support from NSF grant DMS-2424326. The fourth author received support from NSF grants DMS-2101800, DMS-2501903 and Simons Foundation Travel Support for Mathematicians SFI-MPS-TSM00013051 while working on this project. Work on this project was done by all the authors at the BIRS workshop “Notions of Singularity in Different Characteristics” in Banff, Canada, in October 2025.

Footnotes

1 This includes rings essentially of finite type over a field, DVR, or $\mathbb {Z}$ , and F-finite rings of characteristic $p>0$ .

2 By enlarging the residue field, we can assume that J is generated up to integral closure by at most $\dim R$ elements.

3 Note that normalizations are frequently defined only for schemes that locally have finitely many minimal primes.

4 Alternatively, we can also consider the composition $\overline {J^{n+k-1}} \mathcal {O}_Y \to \text {EN}_2 \to \text {EN}_1$ , see Remark 2.1.

5 This can be relaxed in view of [Reference MurayamaMur24] but we restrict to this setting.

6 The alteration topology admits fiber products, denoted $\times ^{\text {alt}}$ , thanks to [Reference Illusie, Laszlo and OrgogozoILO14, Exposé II, Proposition 1.2.6]. It is exactly the dominant component(s) of the usual fiber product. Under this fiber product, $\{{{}^0}Y^j\times _{{{}^0}V}({{}^0}V^{a} \times _{{}^0V}{{}^0}V^b) \to X_0^a \times ^{\text {alt}}_X X_0^b\}$ is a cover in the alteration topology.

References

Aberbach, I. M. and Huneke, C., ‘An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees algebras’, Math. Ann. 297(2) (1993), 343–369.Google Scholar

Aberbach, I. M. and Huneke, C., ‘F-rational rings and the integral closures of ideals’, Michigan Math. J. 49(1) (2001), 3–11.Google Scholar

Aberbach, I. M., Huneke, C., and Trung, N. V., ‘Reduction numbers, Briançon-Skoda theorems and the depth of Rees rings’, Compositio Math. 97(3) (1995), 403–434.Google Scholar

Abhyankar, S. S., Resolution of Singularities of Embedded Algebraic Surfaces, Pure and Applied Mathematics, vol. 24, Academic Press, New York-London, 1966.Google Scholar

Andersson, M., ‘Explicit versions of the Briançon-Skoda theorem with variations’, Michigan Math. J. 54(2) (2006), 361–373.Google Scholar

Andersson, M., Samuelsson, H. k., and Sznajdman, J., ‘On the Briançon-Skoda theorem on a singular variety’, Ann. Inst. Fourier (Grenoble) 60(2) (2010), 417–432.Google Scholar

André, Y., ‘La conjecture du facteur direct’, Publ. Math. Inst. Hautes Études Sci. 127 (2018), 71–93.Google Scholar

Aschenbrenner, M. and Schoutens, H., ‘Lefschetz extensions, tight closure and big Cohen-Macaulay algebras’, Israel J. Math. 161 (2007), 221–310.Google Scholar

Bhatt, B., ‘Derived splinters in positive characteristic’, Compos. Math. 148(6) (2012), 1757–1786.Google Scholar

Bhatt, B., ‘On the direct summand conjecture and its derived variant’, Invent. Math. 212(2) (2018), 297–317.Google Scholar

Bhatt, B., ‘Cohen-Macaulayness of absolute integral closures’, Preprint, 2020, arXiv:2008.08070.Google Scholar

Bhatt, B., Ma, L., Patakfalvi, Z., Schwede, K., Tucker, K., Waldron, J., and Witaszek, J., ‘Perfectoid pure singularities’, Preprint, 2024, arXiv:2409.17965.Google Scholar

Bhatt, B. and Scholze, P., ‘Prisms and prismatic cohomology’, Ann. of Math. (2) 196(3) (2022), 1135–1275.Google Scholar

Brenner, H., ‘How to rescue solid closure’, J. Algebra 265(2) (2003), 579–605.Google Scholar

Buchsbaum, D. A. and Eisenbud, D., ‘Generic free resolutions and a family of generically perfect ideals’, Adv. Math. 18(3) (1975), 245–301.Google Scholar

Christensen, J. D., ‘Ideals in triangulated categories: phantoms, ghosts and skeleta’, Adv. Math. 136(2) (1998), 284–339.Google Scholar

Cid-Ruiz, Y. and Jeffries, J., ‘Uniformity in nonreduced rings via Noetherian operators’, Int. Math. Res. Not. IMRN 17 (2025), Paper No. rnaf260, 8.Google Scholar

Conrad, B., ‘Cohomological descent’, 2003, https://math.stanford.edu/~conrad/papers/hypercover.pdf.Google Scholar

Cossart, V. and Piltant, O., ‘Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings’, J. Algebra 320(3) (2008), 1051–1082.Google Scholar

Cossart, V. and Piltant, O., ‘Resolution of singularities of threefolds in positive characteristic. II’, J. Algebra 321(7) (2009), 1836–1976.Google Scholar

Cossart, V. and Piltant, O., ‘Resolution of singularities of arithmetical threefolds’, J. Algebra 529 (2019), 268–535.Google Scholar

Cutkosky, S. D., ‘Resolution of singularities for 3-folds in positive characteristic’, Amer. J. Math. 131(1) (2009), 59–127.Google Scholar

Datta, R., Private communication, 2025.Google Scholar

Datta, R. and Murayama, T., ‘Permanence properties of F-injectivity’, Math. Res. Lett. 31(4) (2024), 985–1027.Google Scholar

De Deyn, T., Lank, P., Manali-Rahul, K., and Venkatesh, S., ‘Measuring birational derived splinters’, Preprint, 2025, arXiv:2510.26648.Google Scholar

De Fernex, T. and Hacon, C., ‘Singularities on normal varieties’, Compos. Math. 145(2) (2009), 393–414.Google Scholar

Deligne, P., ‘Théorie de Hodge. II’, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.Google Scholar

Du Bois, P., ‘Complexe de de Rham filtré d’une variété singulière’, Bull. Soc. Math. France 109(1) (1981), 41–81.Google Scholar

Duncan, A. J. and O’Carroll, L., ‘A full uniform Artin-Rees theorem’, J. Reine Angew. Math. 394 (1989), 203–207.Google Scholar

Eagon, J. A. and Northcott, D. G., ‘Ideals defined by matrices and a certain complex associated with them’, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204.Google Scholar

Ein, L. and Lazarsfeld, R., ‘A geometric effective Nullstellensatz’, Invent. Math. 137(2) (1999), 427–448.Google Scholar

Eisenbud, D. and Hochster, M., ‘A Nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions’, J. Algebra 58(1) (1979), 157–161.Google Scholar

Epstein, N., McDonald, P. M., Rebecca, R. G., and Schwede, K., ‘Closure operations induced via resolutions of singularities in characteristic zero’, Preprint, 2025, arXiv:2504.05554.Google Scholar

Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry.Google Scholar

Guillén, F., Navarro Aznar, V., Pascual Gainza, P. and Puerta, F., Hyperrésolutions cubiques et descente cohomologique, Lecture Notes in Mathematics, vol. 1335 (Springer-Verlag, Berlin, 1988), Papers from the Seminar on Hodge-Deligne Theory held in Barcelona, 1982.Google Scholar

Hara, N., ‘A characterization of rational singularities in terms of injectivity of Frobenius maps’, Amer. J. Math. 120(5) (1998), 981–996.Google Scholar

Heitmann, R. C., ‘The plus closure in mixed characteristic’, J. Algebra 193(2) (1997), 688–708.Google Scholar

Heitmann, R. C., ‘Extensions of plus closure’, J. Algebra 238(2) (2001), 801–826.Google Scholar

Heitmann, R. C. and Ma, L., ‘Extended plus closure in complete local rings’, J. Algebra 571 (2021), 134–150.Google Scholar

Hironaka, H., ‘Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II’, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326.Google Scholar

Hochster, M. and Huneke, C., ‘Tight closure, invariant theory, and the Briançon-Skoda theorem’, J. Amer. Math. Soc. 3(1) (1990), 31–116.Google Scholar

Hochster, M. and Huneke, C., ‘Infinite integral extensions and big Cohen-Macaulay algebras’, Ann. of Math. (2) 135(1) (1992), 53–89.Google Scholar

Hochster, M. and Huneke, C., ‘Applications of the existence of big Cohen-Macaulay algebras’, Adv. Math. 113(1) (1995), 45–117.Google Scholar

Hochster, M. and Huneke, C., ‘Tight closure in equal characteristic zero’, Preprint, 2006.Google Scholar

Hochster, M. and Roberts, J. L., ‘The purity of the Frobenius and local cohomology’, Adv. Math. 21(2) (1976), 117–172.Google Scholar

Huber, A. and Jörder, C., ‘Differential forms in the h-topology’, Algebr. Geom. 1(4) (2014), 449–478.Google Scholar

Huneke, C., ‘Uniform bounds in Noetherian rings’, Invent. Math. 107(1) (1992), 203–223.Google Scholar

Huneke, C., ‘Desingularizations and the uniform Artin-Rees theorem’, J. Lond. Math. Soc. (2) 62(3) (2000), 740–756.Google Scholar

Huneke, C. and Lyubeznik, G., ‘Absolute integral closure in positive characteristic’, Adv. Math. 210(2) (2007), 498–504.Google Scholar

Huneke, C. and Swanson, I., Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, vol. 336 (Cambridge University Press, Cambridge, 2006).Google Scholar

Huneke, C. and Watanabe, K.-i., ‘Upper bound of multiplicity of F-pure rings’, Proc. Amer. Math. Soc. 143(12) (2015), 5021–5026.Google Scholar

Hyry, E. and O. Villamayor U., ‘A Briançon-Skoda theorem for isolated singularities’, J. Algebra 204(2) (1998), 656–665.Google Scholar

Illusie, L., Laszlo, Y. and Orgogozo, F., ‘Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schéemas quasi-excellents’, Illusie, Luc and Laszlo, Yves and Orgogozo, Fabrice, (Société Mathématique de, Paris, 2014), pp. 363–364.Google Scholar

Katz, D. and Polstra, T., Uniform Briançon-Skoda theorem for non-reduced rings.Google Scholar

Kawasaki, T., ‘On Macaulayfication of Noetherian schemes’, Trans. Amer. Math. Soc. 352(6) (2000), 2517–2552.Google Scholar

Kovács, S. J., ‘A characterization of rational singularities’, Duke Math. J. 102(2) (2000), 187–191.Google Scholar

Kovács, S. J., ‘Non-Cohen-Macaulay canonical singularities’, in Local and global Methods in Algebraic Geometry, Contemp. Math., vol. 712 (Amer. Math. Soc., Providence, RI, 2018), pp. 251–259.Google Scholar

Kovács, S. J., Schwede, K. and Smith, K. E., ‘The canonical sheaf of Du Bois singularities’, Adv. Math. 224(4) (2010), 1618–1640.Google Scholar

Krause, H., Homological Theory of Representations, Cambridge Studies in Advanced Mathematics, vol. 195 (Cambridge University Press, Cambridge, 2022).Google Scholar

Lank, P. and Venkatesh, S., ‘Triangulated characterizations of singularities’, Preprint, 2024, arXiv:2405.04389, to appear in Nagoya Math. J.Google Scholar

Lazarsfeld, R., Positivity in Algebraic Geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49 (Springer-Verlag, Berlin, 2004), Positivity for vector bundles, and multiplier ideals.Google Scholar

Lazarsfeld, R. and Lee, K., ‘Local syzygies of multiplier ideals’, Invent. Math. 167(2) (2007), 409–418.Google Scholar

Lee, B., ‘Local acyclic fibrations and the de Rham complex’, Homology Homotopy Appl. 11(1) (2009), 115–140.Google Scholar

Li, Z., ‘On the Briançon-Skoda theorem for analytic local rings with singularities’, J. Algebra 577 (2021), 45–60.Google Scholar

Lipman, J., ‘Desingularization of two-dimensional schemes’, Ann. of Math. (2) 107(1) (1978), 151–207.Google Scholar

Lipman, J., ‘Adjoints of ideals in regular local rings’, Math. Res. Lett. 1(6) (1994), 739–755, With an appendix by Steven Dale Cutkosky.Google Scholar

Lipman, J. and Sathaye, A., ‘Jacobian ideals and a theorem of Briançon-Skoda’, Michigan Math. J. 28(2) (1981), 199–222.Google Scholar

Lipman, J. and Teissier, B., ‘Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals’, Michigan Math. J. 28(1) (1981), 97–116.Google Scholar

Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford University Press, Oxford, 2002,), Translated from the French by Reinie Erné, Oxford Science Publications.Google Scholar

Lyu, S., ‘On some properties of birational derived splinters’, Preprint, 2022, arXiv:2210.03193, to appear in Michigan Math. J.Google Scholar

Lyu, S., ‘Formal lifting of dualizing complexes and consequences’, Nagoya Math. J. 258 (2025), 219–236.Google Scholar

Ma, L. and Polstra, T., ‘F-singularities: A commutative algebra approach’, available at https://www.math.purdue.edu/~ma326/F-singularitiesBook.pdf, 2025.Google Scholar

Ma, L. and Schwede, K., ‘A Kunz-type characterization of regular rings via alterations’, J. Pure Appl. Algebra 224(3) (2020), 1124–1131.Google Scholar

Ma, L. and Schwede, K., ‘Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras’, Duke Math. J. 170(13) (2021), 2815–2890.Google Scholar

Ma, L., Schwede, K., Tucker, K., Waldron, J. and Witaszek, J., ‘An analogue of adjoint ideals and PLT singularities in mixed characteristic’, J. Algebraic Geom. 31(3) (2022), 497–559.Google Scholar

Mehta, V. B. and Srinivas, V., ‘A characterization of rational singularities’, Asian J. Math. 1(2) (1997), 249–271.Google Scholar

Murayama, T., ‘Injectivity theorems and cubical descent for schemes, stacks, and analytic spaces’, Preprint, 2024, arXiv:2406.10800.Google Scholar

Murayama, T., ‘Relative vanishing theorems for Q-schemes’, Algebr. Geom. 12(1) (2025), 84–144.Google Scholar

O’Carroll, L., ‘A uniform Artin-Rees theorem and Zariski’s main lemma on holomorphic functions’, Invent. Math. 90(3) (1987), 647–652.Google Scholar

Park, S. G., ‘Upper bound on the multiplicity of rational and Du Bois singularities’, Preprint, 2025, arXiv:2509.21807.Google Scholar

Rodríguez-Villalobos, S. and Schwede, K., ‘The Briançon-Skoda Theorem via weak functoriality of big Cohen-Macaulay algebras’, Preprint, 2024, arXiv:2406.02433, to appear in Michigan Math. J.Google Scholar

Schoutens, H., ‘A non-standard proof of the Briançon-Skoda theorem’, Proc. Amer. Math. Soc. 131(1) (2003), 103–112.Google Scholar

Schoutens, H., ‘Non-standard tight closure for affine C-algebras’, Manuscripta Math. 111(3) (2003), 379–412.Google Scholar

SGA, Théorie des topos et cohomologie étale des schémas . Tome 2, Lecture Notes in Mathematics, vol. 270, Springer-Verlag, Berlin-New York, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.Google Scholar

Shibata, K., ‘Upper bound of the multiplicity of a Du Bois singularity’, Proc. Amer. Math. Soc. 145(3) (2017), 1053–1059.Google Scholar

Skoda, H. and Briançon, J., ‘Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de Cn’, C . R. Acad. Sci. Paris Sér. A 278 (1974), 949–951.Google Scholar

Smith, K. E., ‘F-rational rings have rational singularities’, Amer. J. Math. 119(1) (1997), 159–180.Google Scholar

Srinivasan, H., Multiplicative Structures on Some Canonical Resolutions (Resolutions, Algebra Structures, Complexes), ProQuest LLC, Ann Arbor, MI, 1986, Thesis (Ph.D.)–Brandeis University.Google Scholar

Srinivasan, H., ‘Algebra structures on some canonical resolutions’, J. Algebra 122(1) (1989), 150–187.Google Scholar

T. Stacks project authors, ‘The stacks project’, 2025.Google Scholar

Sznajdman, J., ‘An elementary proof of the Briançon-Skoda theorem’, Ann. Fac. Sci. Toulouse Math . (6) 19(3–4) (2010), 675–685.Google Scholar

Sznajdman, J., ‘A Briançon-Skoda-type result for a non-reduced analytic space’, J. Reine Angew. Math. 742 (2018), 1–16.Google Scholar

Tanaka, H., ‘Infinite dimensional excellent rings’, Comm. Algebra 47(2) (2019), 482–489.Google Scholar

Temkin, M., ‘Desingularization of quasi-excellent schemes in characteristic zero’, Adv. Math. 219(2) (2008), 488–522.Google Scholar

Urbinati, S., ‘Discrepancies of non-Q-Gorenstein varieties’, Michigan Math. J. 61(2) (2012), 265–277.Google Scholar

Wang, H.-J., ‘Some uniform properties of 2-dimensional local rings’, J. Algebra 188(1) (1997), 1–15.Google Scholar

Wheeler, A. and Zhang, W., ‘Remarks on effective uniform Briançon-Skoda’, Preprint, 2025, arXiv:2510.04004.Google Scholar

Zhou, C., ‘Uniform annihilators of local cohomology of excellent rings’, J. Algebra 315(1) (2007), 286–300.Google Scholar

Article contents

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

Abstract

MSC classification

Information

1 Introduction

Main Theorem (Theorem 2.2)

Corollary A (Theorem 3.12)

Main Application (Corollary 5.6, Corollary 5.4, cf. [Reference HunekeHun92, Conjectures 1.3 and 1.4])

Corollary B (Theorem 3.18)

Corollary C (Proposition 4.10, Proposition 4.11)

1.1 History of this paper and strategy of the proof

2 Integral closure and the Buchsbaum-Eisenbud complex

2.1 Partially normalized blowups and Čech covers

2.2 The Buchsbaum-Eisenbud complex

2.3 The main result

3 Generalizations of rational and Du Bois singularities

Definition 3.1 [Reference Lipman and TeissierLT81, Section 2]

Definition 3.2 [Reference ChristensenChr98],[Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMP+24, Proposition 2.10]

Lemma 3.3 (cf. [Reference LyuLyu22, Proposition 3.10])

Definition 3.5 [Reference LyuLyu22]

3.1 Briançon-Skoda theorem for birational derived splinters

3.2 A characteristic-free weakening of Du Bois singularities

Proof of claim

3.3 Briançon-Skoda theorem for perfectoid rings

4 Connections with closure operations

4.1 Characteristic zero

4.2 Characteristic $p> 0$

Proposition 4.10 [Reference Hochster and HunekeHH95, Theorem 7.1]

4.3 Mixed characteristic

Proposition 4.11 [Reference Rodríguez-Villalobos and SchwedeRS24, Reference HeitmannHei01]

5 Uniform Briançon-Skoda and uniform Artin-Rees

Corollary 5.4 (Uniform Briançon-Skoda)

Corollary 5.6 (Uniform Artin-Rees)

Acknowledgments

Competing interests

Funding statement

Footnotes

References

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