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The Brian\c{c}on-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings

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Pith's one-line read The paper proves the full Briançon-Skoda containment \overline{J^{n+k-1}}\subseteq J^k for birational derived splinters—including pseudo-rational rings—and derives uniform Briançon-Skoda and Artin-Rees bounds for quasi-excellent finite-dime

desk verdict The core theorem is elegant and sound; the uniform Section 5 has a repairable but real gap. read the letter →

arxiv 2510.11540 v4 pith:SQL477JI submitted 2025-10-13 math.AC math.AG

Linquan Ma , Peter M. McDonald , Rebecca R.G. , Karl Schwede This is my paper · ORCID
classification math.ACmath.AG MSC 13A1513B2213D0213D4514B0513A35
keywords Briançon-Skodatheoremintegralclosurepseudo-rationalsingularitiesbirationalderivedsplintersBuchsbaum-EisenbudcomplexuniformArtin-Reesexcellentrings
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the classical Briançon-Skoda containment of integral closures, $\overline{J^{n+k-1}}\subseteq J^k$, holds in full strength for a broad class of singular rings: birational derived splinters, which include pseudo-rational rings and characteristic-p analogs such as F-rational rings, with a slightly weakened version for Du Bois-type singularities. The proof is short and derived-categorical: it shows that the relevant ideal sheaf is zero in a derived tensor product with the Buchsbaum-Eisenbud complex on a blowup, requiring no vanishing theorems or Cohen-Macaulayness. If correct, the same mechanism also yields the uniform Briançon-Skoda and uniform Artin-Rees theorems for quasi-excellent reduced (respectively, all) Noetherian rings of finite dimension, resolving long-standing uniformity conjectures. The paper also recovers and unifies tight-closure, plus-closure, and mixed-characteristic closure versions of Briançon-Skoda.

What carries the argument

The load-bearing object is the Buchsbaum-Eisenbud complex $L_k(f)$ associated to the $k$-th power of $(f_1,\dots,f_n)$—a free complex whose zeroth homology is $R/J^k$ and which resolves $R/J^k$ when the $f_i$ form a regular sequence; it is isomorphic to a specialization of the Eagon-Northcott complex. The paper constructs an exact subcomplex BE2 on the blowup $Y$ of $J^{n+k-1}$, twisted so that its final term is $O_Y(-(n+k-1)E)$, and compares it with the pullback of $L_k(f)$. Because BE2 is exact, the map between them is zero in the derived category, giving the main vanishing statement. The uniform results additionally use weak local uniformization to build alteration hypercovers with regular terms, on which th

What would settle it

Find a finite-dimensional quasi-excellent domain $R$ for which the intersection $T_d(R)=\bigcap_{I,n}(I^{n-d}:I^n)$ is zero, or exhibit a truncated regular hypercover of the kind in Lemma 5.1 that admits no map to a full regular alteration hypercover; either would block the uniformity proof. A more direct check: compute $T_d(R)$ for a concrete excellent ring and see whether it contains a nonzero element.

Watch

Extended reading notes

Core claim

The central claim, on the paper's own terms, is Theorem 2.2: for any ring $R$ and any $n$-generated ideal $J=(f_1,\dots,f_n)$, if $Y$ is the blowup of $J^{n+k-1}$ (or any map dominating it), then the canonical map $O_Y(-(n+k-1)E)\to L_k(f)\otimes O_Y$ is zero in the derived category, where $L_k(f)$ is the Buchsbaum-Eisenbud complex of the $k$-th power of $J$. Taking zeroth cohomology, $J^{n+k-1}$ maps to zero in $H^0(L_k(f)\otimes^\mathbf{L} R\Gamma(Y,\mathcal{O}_Y))$; since the zeroth homology of $L_k(f)$ is $R/J^k$, this gives a containment of the integral closure in the kernel of the natural map. When $R$ is a birational derived splinter—every pseudo-rational ring is one, by an argument of Kovács—that kernel is exac

Load-bearing premise

The uniform half depends on the claim that every $n$-truncated hypercover of an excellent finite-dimensional domain by regular schemes can be extended to a full alteration hypercover whose terms are all regular, a step the paper justifies by invoking weak local uniformization but does not construct, and whose compatibility it explicitly concedes may fail.

Editorial extensions

If this is right

  • For every n-generated ideal J in a birational derived splinter (e.g., pseudo-rational, F-rational, BCM-rational, or +-rational), the full Briançon-Skoda containment \overline{J^{n+k-1}}\subseteq J^k holds for all k.
  • For reduced blowup-square splinters—including Du Bois, F-pure, and Cohen-Macaulay F-injective singularities—the containment \overline{J^{n+k}}\subseteq J^k holds.
  • Quasi-excellent reduced rings of finite dimension satisfy a uniform Briançon-Skoda theorem: a single integer k works for all ideals I, giving I^n\subseteq I^{n-k} for all n\geq k.
  • Quasi-excellent rings of finite dimension satisfy a uniform Artin-Rees theorem: for each pair of finitely generated modules N\subseteq M, one integer \ell works for all ideals I, giving I^nM\cap N\subseteq I^{n-\ell}M for all n\geq \ell.
  • The theorem recovers and unifies closure-based Briançon-Skoda results, including plus closure and tight closure in characteristic p>0 and their mixed-characteristic analogues.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the vanishing already occurs on Y, before taking cohomology, which suggests the method may transfer to other bases or non-Noetherian settings; the paper itself applies it to perfectoid rings, where it yields J^{n+k}\subseteq J^k.
  • Editorial extension: tracking the degrees and ranks in the Buchsbaum-Eisenbud comparison could make the uniform Artin-Rees constant effective rather than existential, since the complex is explicit.
  • Editorial extension: the paper's proof of the uniform results depends on extending truncated regular hypercovers, a step it does not construct; if that extension is repaired, the dimension parameter d in T_d(R) might be lowered to the analytic spread or minimal number of generators.
  • Editorial extension: the same exact-complex argument might yield analogous containments for other closure operations defined by resolution-like objects, since the birational pre-closure introduced here is shown to dominate many standard closures.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper proves a derived-category Briançon–Skoda type statement (Theorem 2.2): for an n-generated ideal J=(f_1,...,f_n) in any ring R, if Y is the blowup of J^{n+k-1}, then the canonical map O_Y(-(n+k-1)E) → L_k(f) ⊗ O_Y is zero in D(Y), where L_k(f) is the Buchsbaum–Eisenbud/Eagon–Northcott complex. Taking cohomology and using splitting hypotheses yields J^{n+k-1} ⊆ J^k for birational derived splinters (e.g. pseudo-rational rings, F-rational/BCM-rational/+rational excellent rings) and J^{n+k} ⊆ J^k for reduced blowup-square splinters (e.g. Du Bois, lim-perfectoid pure, CM lim-perfectoid injective rings). Sections 4 derives closure-operation versions (tight, plus, ep closures). Section 5 aims to prove Huneke's uniform Briançon–Skoda and uniform Artin–Rees conjectures for quasi-excellent finite-dimensional rings, via a nonvanishing statement T_d(R) ≠ 0 whose proof relies on an alteration-hypercover lemma (Lemma 5.1).

Significance. Theorem 2.2 is a genuinely elegant and short result, and Corollaries A and B are substantial advances: they give the full Briançon–Skoda containment for pseudo-rational singularities in all characteristics and a characteristic-free weakening of Du Bois singularities. The closure-operation corollaries unify known results. If the uniformity arguments in Section 5 were correct, the paper would also resolve Huneke's conjectures on uniform Briançon–Skoda and uniform Artin–Rees. However, the uniformity section currently contains serious gaps, one of which is explicitly acknowledged by the authors in the proof of Lemma 5.1 and another that appears to be a genuine error in the passage from a minimal reduction to the original ideal in Theorem 5.2. The non-uniform portions of the paper are likely correct and valuable, but the advertised uniform conjectures are not established as written.

major comments (3)
  1. [§5, Theorem 5.2 (also §5, initial outline)] The proof of Theorem 5.2 asserts that, for a minimal reduction J of an ideal I with at most d+1 generators, the natural map I^{d+k}→R→RΓ(Y,O_Y)→L_k(J)⊗^L RΓ(Y,O_Y) is zero, citing Theorem 2.2. However, Theorem 2.2 applies to the ideal being blown up, namely J, and gives vanishing for J^{d+k}, not for I^{d+k}. Since J⊆I, there is no general containment I^{d+k}⊆J^{d+k}; for example I=(x,y) in k[x,y] and J=(x). The subsequent conclusion c I^{d+k}⊆J^k therefore does not follow. This is a load-bearing step for the proof of T_d(R)≠0 and for Corollaries 5.3 and 5.5.
  2. [§5, Lemma 5.1] The statement of Lemma 5.1 claims a commutative diagram involving a vertical map nV●→X● between the Zariski hypercover nV● and the alteration hypercover X● with regular terms. The final paragraph of the proof explicitly disclaims such a map: 'there might not be a map from nV●→X● fitting the commutative diagram above'. This is an internal contradiction in the statement of the lemma. Since Theorem 5.2 invokes this lemma with n=d+2 and Corollaries 5.3/5.5 depend on Theorem 5.2, the uniform results are not supported. The lemma must either be proved with the full map, or the statement and the proof of Theorem 5.2 must be revised to show that only the truncated diagram is needed, with a careful justification of the descent step.
  3. [§5, proof of Theorem 5.2, descent from X● to V≤d+2] Even if Lemma 5.1 is repaired, the step asserting that zero-ness of the map to L_k(J)⊗^L RΓ(X●) implies zero-ness of the composition to L_k(J)⊗^L RΓ(V≤d+2) needs a detailed verification. The text writes '→L_k(J)⊗^L RΓ(X≤d+2)→L_k(J)⊗^L RΓ(V≤d+2)' and says 'the diagram guarantees' the vanishing, but the functoriality of truncations and the compatibility of the maps R→RΓ(X●) and R→RΓ(V≤d+2) are not spelled out. In the absence of a full map V●→X●, it is not automatic that the zero map factors through the truncated diagram in the claimed way. This is a technical but essential gap.
minor comments (3)
  1. [Abstract, acknowledgements] There are typos: 'Brian\c{c}on-Skoda' appears as a raw LaTeX command in the abstract, and 'coversations' in the acknowledgements should be 'conversations'.
  2. [§3, Theorem 3.17, last paragraph] The exponent in O_Y(−(n−k−1)E−F) appears to be a typo; it should presumably be O_Y(−(n+k−1)E−F) to match the surrounding argument and the desired containment Γ(...)⊆J^k.
  3. [Abstract vs. §5] The abstract says the uniform theorems hold for 'excellent' rings, while Corollaries 5.3 and 5.5 state 'quasi-excellent' rings. This inconsistency should be fixed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is a direct complex computation on a blowup; self-citations are contextual and not load-bearing, and the flagged Lemma 5.1 caveat is a proof gap rather than a circular step.

full rationale

The paper's main result (Theorem 2.2) is derived by constructing an exact Buchsbaum–Eisenbud complex BE2 on the blowup and observing that the canonical map from O_Y(-(n+k-1)E) into L_k(f)⊗O_Y factors through an exact complex: "Since BE2 is exact and hence zero in the derived category, O_Y(−(n+k−1)E)→BE_1 = L_k(f)⊗O_Y is zero in the derived category." This is a direct computation, not an assumption of the target containment. The subsequent Briançon–Skoda statements for birational derived splinters and Du Bois-type singularities use this theorem together with splitting or purity properties that are independently established or cited from external work; no fitted parameter is renamed as a prediction. Self-citations such as [EMRS25], [Lyu22], and [MP25] are used for definitions, context, and examples rather than to import the conclusions. The uniform results in Section 5 rely on Lemma 5.1, whose proof ends with an explicit caveat: "Note that there might not be a map from nV●→X● fitting the commutative diagram above though." This is an internal proof gap or unproven compatibility claim about hypercovers, not a circular reduction: Lemma 5.1 does not assume the Briançon–Skoda containment or any equivalent thereof. Theorem 5.2 then uses the asserted commutative diagram to descend a splitting; if the diagram fails, the uniform Huneke-conjecture results are unsupported. That is a correctness risk, not a form of circularity under the criteria of this pass. The paper is self-contained against the external benchmarks of Theorem 2.2 and its non-uniform consequences, and no step reduces by definition or by self-citation to the conclusion.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

The paper is parameter-free; it does not fit constants to data. It rests on a stable of deep external theorems (Gabber, Zhou, Huneke, Bhatt, Lyu), listed above, and introduces two new definitions used as tools. No physical or empirical entities are posited.

assumptions (8)
  • domain assumption Gabber's weak local uniformization: every excellent scheme admits covers by regular schemes in the alteration topology (ILO14, Exp. VII, Thm 1.1).
    Entry point for Lemma 5.1; if only weaker uniformization held, the regular hypercover X● used in Theorem 5.2 might not exist.
  • domain assumption Zhou [Zho07]: CM(R) ≠ 0 for excellent rings of finite dimension.
    Combined with T(R/P)≠0 in Huneke's criterion in Corollaries 5.3 and 5.5.
  • standard math Huneke [Hun92, Thm 3.4 / Prop 3.7]: uniform Artin–Rees and uniform Briançon–Skoda follow from T(R/P)≠0 and CM(R/P)≠0.
    The bridge from Theorem 5.2 to Corollaries 5.3 and 5.5.
  • domain assumption Lyu [Lyu25, Cor 6.6]: quasi-excellent rings can be replaced by excellent rings after a faithfully flat étale base change.
    Used to extend the excellent-ring arguments to quasi-excellent rings in Corollaries 5.3 and 5.5.
  • domain assumption Bhatt [Bha12, Bha20] factorizations R→RΓ(Y,O_Y)→R^+ in characteristic p and R→RΓ(Y,O_Y)→\hat{R}^+ in mixed characteristic.
    Converts the birational pre-closure into plus closure and extended plus closure in Propositions 4.9 and 4.10.
  • domain assumption Smith [Smi97] and Ma–Schwede [MS21]: F-rational, BCM-rational, and +-rational excellent local rings are pseudo-rational.
    Lets Corollary A cover these classes via Lemma 3.8.
  • domain assumption Du Bois / perfectoid purity results: Du Bois ⇒ reduced blowup-square splinter via the Deligne–Du Bois complex; lim-perfectoid pure and lim-perfectoid injective rings are reduced blowup-square splinters ([BMP+24]).
    Used in Proposition 3.15 to reach the Du Bois and perfectoid BS statements.
  • standard math Huneke–Swanson [HS06] facts on minimal reductions: in a local ring with infinite residue field, an ideal with analytic spread ≤n has an n-generated minimal reduction preserving integral closures of powers.
    Reduces BS statements for analytic spread ≤n to n-generated ideals in Theorems 3.11 and 3.17.
invented entities (2)
  • birational pre-closure J^Bir independent evidence
    purpose: Unify closure-based Briançon–Skoda statements by taking kernels of maps to R/J^k ⊗^L RΓ(Y,O_Y) over all proper birational Y.
    Defined in §4; it is a concrete ideal whose inclusions J^{n+k-1}⊆(J^k)^Bir and I^Bir⊆I^+ (or \hat{R}^+∩R) can be checked on explicit rings and recover known closure operations.
  • reduced blowup-square splinter independent evidence
    purpose: A characteristic-free singularity class weaker than Du Bois, F-pure, and F-injective that satisfies the BS-style bound J^{n+k}⊆J^k.
    Defined in §3.2; Proposition 3.15 identifies Du Bois, lim-perfectoid pure, and Cohen-Macaulay lim-perfectoid injective rings as examples, and Corollary 3.18 gives testable multiplicity bounds.

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Pith. "Pith review of The Brian\c{c}on-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings." pith.science (2026). https://pith.science/paper/SQL477JI

@misc{pith2026251011540,
  author       = {Pith},
  title        = {Pith review of: The Brian\ccon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQL477JI}},
  note         = {Machine review of arXiv:2510.11540}
}
abstract

Suppose $J = (f_1, \dots, f_n)$ is an $n$-generated ideal in any ring $R$. We prove a general Brian\c{c}on-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 Brian\c{c}on-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 Brian\c{c}on-Skoda-type theorem also implies well-known closure-based Brian\c{c}on-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 Brian\c{c}on-Skoda theorem for quasi-excellent, respectively quasi-excellent reduced, rings of finite dimension, answering conjectures of Huneke.

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Forward citations

Cited by 2 Pith papers

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  1. Factoring maps to big Cohen-Macaulay algebras through blowups

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    Functorial balanced big Cohen-Macaulay algebra assignments factor through RΓ(Y, O_Y) for every proper birational map Y → Spec R.

  2. Measuring birational derived splinters

    math.AG 2025-10 accept novelty 6.0 of 10

    The paper defines μ_bds, a level-based invariant of the derived category that measures the failure of a scheme to be a birational derived splinter.

Reference graph

Works this paper leans on

80 extracted references · 5 linked inside Pith · cited by 2 Pith papers

  1. [1]

    I. M. Aberbach and C. Huneke : An improved B rian con- S koda theorem with applications to the C ohen- M acaulayness of R ees algebras , Math. Ann. 297 (1993), no. 2, 343--369. 1241812

    1993

  2. [2]

    I. M. Aberbach and C. Huneke : F-rational rings and the integral closures of ideals, Michigan Math. J. 49 (2001), no. 1, 3--11. 1827071

    2001

  3. [3]

    I. M. Aberbach, C. Huneke, and N. V. Trung : Reduction numbers, B rian con- S koda theorems and the depth of R ees rings , Compositio Math. 97 (1995), no. 3, 403--434. 1353282

    1995

  4. [4]

    S. S. Abhyankar : Resolution of singularities of embedded algebraic surfaces, Pure and Applied Mathematics, vol. Vol. 24, Academic Press, New York-London, 1966. 217069

    1966

  5. [5]

    Andersson : Explicit versions of the B rian con- S koda theorem with variations , Michigan Math

    M. Andersson : Explicit versions of the B rian con- S koda theorem with variations , Michigan Math. J. 54 (2006), no. 2, 361--373. 2252765

    2006

  6. [6]

    Andersson, H

    M. Andersson, H. k. Samuelsson, and J. Sznajdman : On the B rian con- S koda theorem on a singular variety , Ann. Inst. Fourier (Grenoble) 60 (2010), no. 2, 417--432. 2667781

    2010

  7. [7]

    Andr\'e : La conjecture du facteur direct, Publ

    Y. Andr\'e : La conjecture du facteur direct, Publ. Math. Inst. Hautes \'Etudes Sci. 127 (2018), 71--93. 3814651

    2018

  8. [8]

    Aschenbrenner and H

    M. Aschenbrenner and H. Schoutens : Lefschetz extensions, tight closure and big C ohen- M acaulay algebras , Israel J. Math. 161 (2007), 221--310. 2350164

    2007

Show all 80 references
  1. [9]

    Bhatt : Derived splinters in positive characteristic, Compos

    B. Bhatt : Derived splinters in positive characteristic, Compos. Math. 148 (2012), no. 6, 1757--1786. 2999303

    2012

  2. [10]

    Bhatt : On the direct summand conjecture and its derived variant, Invent

    B. Bhatt : On the direct summand conjecture and its derived variant, Invent. Math. 212 (2018), no. 2, 297--317. 3787829

    2018

  3. [11]

    Bhatt : Cohen-Macaulayness of absolute integral closures , arXiv e-prints (2020), arXiv:2008.08070

    B. Bhatt : Cohen-Macaulayness of absolute integral closures , arXiv e-prints (2020), arXiv:2008.08070

    2020 arXiv

  4. [12]

    Bhatt , L

    B. Bhatt , L. Ma , Z. Patakfalvi , K. Schwede , K. Tucker , J. Waldron , and J. Witaszek : Perfectoid pure singularities , arXiv e-prints (2024), arXiv:2409.17965

    2024 arXiv

  5. [13]

    Bhatt and P

    B. Bhatt and P. Scholze : Prisms and prismatic cohomology, Ann. of Math. (2) 196 (2022), no. 3, 1135--1275. 4502597

    2022

  6. [14]

    Brenner : How to rescue solid closure, J

    H. Brenner : How to rescue solid closure, J. Algebra 265 (2003), no. 2, 579--605. 1987018

    2003

  7. [15]

    D. A. Buchsbaum and D. Eisenbud : Generic free resolutions and a family of generically perfect ideals, Advances in Math. 18 (1975), no. 3, 245--301. 396528

    1975

  8. [16]

    J. D. Christensen : Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998), no. 2, 284--339. 1626856

    1998

  9. [17]

    Cossart and O

    V. Cossart and O. Piltant : Resolution of singularities of threefolds in positive characteristic. I . R eduction to local uniformization on A rtin- S chreier and purely inseparable coverings , J. Algebra 320 (2008), no. 3, 1051--1082. 2427629

    2008

  10. [18]

    Cossart and O

    V. Cossart and O. Piltant : Resolution of singularities of threefolds in positive characteristic. II , J. Algebra 321 (2009), no. 7, 1836--1976. 2494751

    2009

  11. [19]

    Cossart and O

    V. Cossart and O. Piltant : Resolution of singularities of arithmetical threefolds, J. Algebra 529 (2019), 268--535. 3942183

    2019

  12. [20]

    S. D. Cutkosky : Resolution of singularities for 3-folds in positive characteristic, Amer. J. Math. 131 (2009), no. 1, 59--127. 2488485

    2009

  13. [21]

    Datta : Private communication, 2025

    R. Datta : Private communication, 2025

    2025

  14. [22]

    Datta and T

    R. Datta and T. Murayama : Permanence properties of F -injectivity , Math. Res. Lett. 31 (2024), no. 4, 985--1027. 4831046

    2024

  15. [23]

    De Deyn , P

    T. De Deyn , P. Lank, K. Manali-Rahul , and S. Venkatesh : Measuring birational derived splinters

  16. [24]

    De Fernex and C

    T. De Fernex and C. Hacon : Singularities on normal varieties, Compos. Math. 145 (2009), no. 2, 393--414

    2009

  17. [25]

    Du Bois : Complexe de de R ham filtr\'e d'une vari\'et\'e singuli\`ere , Bull

    P. Du Bois : Complexe de de R ham filtr\'e d'une vari\'et\'e singuli\`ere , Bull. Soc. Math. France 109 (1981), no. 1, 41--81. MR613848 (82j:14006)

    1981

  18. [26]

    J. A. Eagon and D. G. Northcott : Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188--204. 142592

    1962

  19. [27]

    Ein and R

    L. Ein and R. Lazarsfeld : A geometric effective N ullstellensatz , Invent. Math. 137 (1999), no. 2, 427--448. 1705839

    1999

  20. [28]

    Epstein , P

    N. Epstein , P. M. McDonald , G. Rebecca R. , and K. Schwede : Closure operations induced via resolutions of singularities in characteristic zero , arXiv e-prints (2025), arXiv:2504.05554

    2025

  21. [29]

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

  22. [30]

    , N avarro Aznar, V

    G uill \'e n, F. , N avarro Aznar, V. , P ascual Gainza, P. , and P uerta, F. : Hyperr\'esolutions 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, 198...

    1988

  23. [31]

    Hara : A characterization of rational singularities in terms of injectivity of F robenius maps , Amer

    N. Hara : A characterization of rational singularities in terms of injectivity of F robenius maps , Amer. J. Math. 120 (1998), no. 5, 981--996. MR1646049 (99h:13005)

    1998

  24. [32]

    R. C. Heitmann : The plus closure in mixed characteristic, J. Algebra 193 (1997), no. 2, 688--708. 1458810

    1997

  25. [33]

    R. C. Heitmann : Extensions of plus closure, J. Algebra 238 (2001), no. 2, 801--826. 1823785

    2001

  26. [34]

    R. C. Heitmann and L. Ma : Extended plus closure in complete local rings, J. Algebra 571 (2021), 134--150. 4200713

    2021

  27. [35]

    Hironaka : Resolution of singularities of an algebraic variety over a field of characteristic zero

    H. Hironaka : 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. MR0199184 (33 \#7333)

    1964

  28. [36]

    Hochster and C

    M. Hochster and C. Huneke : Tight closure, invariant theory, and the B rian c on- S koda theorem , J. Amer. Math. Soc. 3 (1990), no. 1, 31--116. 1017784

    1990

  29. [37]

    Hochster and C

    M. Hochster and C. Huneke : Infinite integral extensions and big C ohen- M acaulay algebras , Ann. of Math. (2) 135 (1992), no. 1, 53--89. 1147957

    1992

  30. [38]

    Hochster and C

    M. Hochster and C. Huneke : Applications of the existence of big C ohen- M acaulay algebras , Adv. Math. 113 (1995), no. 1, 45--117. 1332808

    1995

  31. [39]

    Hochster and C

    M. Hochster and C. Huneke : Tight closure in equal characteristic zero, A preprint of a manuscript, 2006

    2006

  32. [40]

    Hochster and J

    M. Hochster and J. L. Roberts : The purity of the F robenius and local cohomology , Advances in Math. 21 (1976), no. 2, 117--172. 417172

    1976

  33. [41]

    Huber and C

    A. Huber and C. J\"order : Differential forms in the h-topology, Algebr. Geom. 1 (2014), no. 4, 449--478. 3272910

    2014

  34. [42]

    Huneke : Uniform bounds in N oetherian rings , Invent

    C. Huneke : Uniform bounds in N oetherian rings , Invent. Math. 107 (1992), no. 1, 203--223. 1135470

    1992

  35. [43]

    Huneke : Desingularizations and the uniform A rtin- R ees theorem , J

    C. Huneke : Desingularizations and the uniform A rtin- R ees theorem , J. London Math. Soc. (2) 62 (2000), no. 3, 740--756. 1794281

    2000

  36. [44]

    Huneke and G

    C. Huneke and G. Lyubeznik : Absolute integral closure in positive characteristic, Adv. Math. 210 (2007), no. 2, 498--504. 2303230

    2007

  37. [45]

    Huneke and I

    C. Huneke and I. Swanson : Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006

    2006

  38. [46]

    Huneke and K.-i

    C. Huneke and K.-i. Watanabe : Upper bound of multiplicity of F -pure rings , Proc. Amer. Math. Soc. 143 (2015), no. 12, 5021--5026. 3411123

    2015

  39. [47]

    Hyry and O

    E. Hyry and O. Villamayor U. : A B rian con- S koda theorem for isolated singularities , J. Algebra 204 (1998), no. 2, 656--665. 1624420

    1998

  40. [48]

    S. J. Kov \'a cs : A characterization of rational singularities, Duke Math. J. 102 (2000), no. 2, 187--191. MR1749436 (2002b:14005)

    2000

  41. [49]

    S. J. Kov\'acs : Non- C ohen- M acaulay canonical singularities , Local and global methods in algebraic geometry, Contemp. Math., vol. 712, Amer. Math. Soc., [Providence], RI, [2018] 2018, pp. 251--259. 3832406

    2018

  42. [50]

    S. J. Kov \'a cs, K. Schwede, and K. E. Smith : The canonical sheaf of D u B ois singularities , Adv. Math. 224 (2010), no. 4, 1618--1640. 2646306

    2010

  43. [51]

    Krause : Homological theory of representations, Cambridge Studies in Advanced Mathematics, vol

    H. Krause : Homological theory of representations, Cambridge Studies in Advanced Mathematics, vol. 195, Cambridge University Press, Cambridge, 2022. 4327095

    2022

  44. [52]

    Lank and S

    P. Lank and S. Venkatesh : Triangulated categories of singularities, Nagoya Mathematical Journal (2025), 1–15

    2025

  45. [53]

    Lazarsfeld and K

    R. Lazarsfeld and K. Lee : Local syzygies of multiplier ideals, Invent. Math. 167 (2007), no. 2, 409--418. 2270459

    2007

  46. [54]

    Li : On the B rian con- S koda theorem for analytic local rings with singularities , J

    Z. Li : On the B rian con- S koda theorem for analytic local rings with singularities , J. Algebra 577 (2021), 45--60. 4232633

    2021

  47. [55]

    Lipman : Desingularization of two-dimensional schemes, Ann

    J. Lipman : Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151--207. 491722

    1978

  48. [56]

    Lipman : Adjoints of ideals in regular local rings, Math

    J. Lipman : Adjoints of ideals in regular local rings, Math. Res. Lett. 1 (1994), no. 6, 739--755, With an appendix by Steven Dale Cutkosky

    1994

  49. [57]

    Lipman and A

    J. Lipman and A. Sathaye : Jacobian ideals and a theorem of B rian c on- S koda , Michigan Math. J. 28 (1981), no. 2, 199--222. 616270

    1981

  50. [58]

    Lipman and B

    J. Lipman and B. Teissier : Pseudorational local rings and a theorem of B rian con- S koda about integral closures of ideals , Michigan Math. J. 28 (1981), no. 1, 97--116. MR600418 (82f:14004)

    1981

  51. [59]

    Liu : Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol

    Q. Liu : Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002, Translated from the French by Reinie Ern \'e , Oxford Science Publications. 1917232

    2002

  52. [60]

    Lyu : On some properties of birational derived splinters , arXiv e-prints (2022), arXiv:2210.03193, to appear in Michigan Math

    S. Lyu : On some properties of birational derived splinters , arXiv e-prints (2022), arXiv:2210.03193, to appear in Michigan Math. J

    2022 arXiv

  53. [61]

    Ma and T

    L. Ma and T. Polstra : F-singularities: A commutative algebra approach, a draft of a book, https://www.math.purdue.edu/ ma326/F-singularitiesBook.pdf, 2025

    2025

  54. [62]

    Ma and K

    L. Ma and K. Schwede : Singularities in mixed characteristic via perfectoid big C ohen- M acaulay algebras , Duke Math. J. 170 (2021), no. 13, 2815--2890. 4312190

    2021

  55. [63]

    L. Ma, K. Schwede, K. Tucker, J. Waldron, and J. Witaszek : An analogue of adjoint ideals and PLT singularities in mixed characteristic , J. Algebraic Geom. 31 (2022), no. 3, 497--559. 4484548

    2022

  56. [64]

    V. B. Mehta and V. Srinivas : A characterization of rational singularities, Asian J. Math. 1 (1997), no. 2, 249--271. MR1491985 (99e:13009)

    1997

  57. [65]

    Murayama : Injectivity theorems and cubical descent for schemes, stacks, and analytic spaces , arXiv e-prints (2024), arXiv:2406.10800

    T. Murayama : Injectivity theorems and cubical descent for schemes, stacks, and analytic spaces , arXiv e-prints (2024), arXiv:2406.10800

    2024 arXiv

  58. [66]

    Murayama : Relative vanishing theorems for Q -schemes , Algebr

    T. Murayama : Relative vanishing theorems for Q -schemes , Algebr. Geom. 12 (2025), no. 1, 84--144. 4841227

    2025

  59. [67]

    S. G. Park : Upper bound on the multiplicity of rational and Du Bois singularities , arXiv e-prints (2025), arXiv:2509.21807

    2025

  60. [68]

    Rodr \' guez-Villalobos and K

    S. Rodr \' guez-Villalobos and K. Schwede : The Brian c on-Skoda Theorem via weak functoriality of big Cohen-Macaulay algebras , arXiv e-prints (2024), arXiv:2406.02433, to appear in Michigan Math. J

    2024 arXiv

  61. [69]

    Schoutens : A non-standard proof of the B rian con- S koda theorem , Proc

    H. Schoutens : A non-standard proof of the B rian con- S koda theorem , Proc. Amer. Math. Soc. 131 (2003), no. 1, 103--112. 1929029

    2003

  62. [70]

    Schoutens : Non-standard tight closure for affine C -algebras , Manuscripta Math

    H. Schoutens : Non-standard tight closure for affine C -algebras , Manuscripta Math. 111 (2003), no. 3, 379--412. 1993501

    2003

  63. [71]

    Shibata : Upper bound of the multiplicity of a D u B ois singularity , Proc

    K. Shibata : Upper bound of the multiplicity of a D u B ois singularity , Proc. Amer. Math. Soc. 145 (2017), no. 3, 1053--1059. 3589305

    2017

  64. [72]

    Skoda and J

    H. Skoda and J. Brian c on : Sur la cl\^ o ture int\' e grale d'un id\' e al de germes de fonctions holomorphes en un point de C n , C. R. Acad. Sci. Paris S\' e r. A 278 (1974), 949--951. 340642

    1974

  65. [73]

    K. E. Smith : F -rational rings have rational singularities , Amer. J. Math. 119 (1997), no. 1, 159--180. 1428062

    1997

  66. [74]

    H. Srinivasan : M ULTIPLICATIVE STRUCTURES ON SOME CANONICAL RESOLUTIONS ( RESOLUTIONS , ALGEBRA STRUCTURES , COMPLEXES ) , ProQuest LLC, Ann Arbor, MI, 1986, Thesis (Ph.D.)--Brandeis University. 2635041

    1986

  67. [75]

    Srinivasan : Algebra structures on some canonical resolutions, J

    H. Srinivasan : Algebra structures on some canonical resolutions, J. Algebra 122 (1989), no. 1, 150--187. 994942

    1989

  68. [76]

    Stacks project authors : The stacks project, 2025

    T. Stacks project authors : The stacks project, 2025

    2025

  69. [77]

    Sznajdman : An elementary proof of the B rian con- S koda theorem , Ann

    J. Sznajdman : An elementary proof of the B rian con- S koda theorem , Ann. Fac. Sci. Toulouse Math. (6) 19 (2010), no. 3-4, 675--685. 2790814

    2010

  70. [78]

    Temkin : Desingularization of quasi-excellent schemes in characteristic zero, Adv

    M. Temkin : Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math. 219 (2008), no. 2, 488--522. 2435647

    2008

  71. [79]

    Urbinati : Discrepancies of non- Q - G orenstein varieties , Michigan Math

    S. Urbinati : Discrepancies of non- Q - G orenstein varieties , Michigan Math. J. 61 (2012), no. 2, 265--277. 2944480

    2012

  72. [80]

    Wheeler and W

    A. Wheeler and W. Zhang : Remarks on effective uniform Brian c on-Skoda , arXiv e-prints (2025), arXiv:2510.04004

    2025

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