{"id":"8238672e-5ca2-4a9e-91ef-f167443c7832","arxiv_id":"2510.11540","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For pseudo-rational and many Du Bois singularities, the full Briançon–Skoda containment J^{n+k-1} ⊆ J^k holds, and quasi-excellent finite-dimensional rings satisfy uniform Briançon–Skoda and uniform Artin–Rees.","lead":"This paper proves new containment theorems for integral closures of powers of ideals, showing the Briançon–Skoda bound holds for pseudo-rational and Du Bois singularities. It also settles two of Huneke's conjectures by proving uniform Briançon–Skoda and uniform Artin–Rees for quasi-excellent rings of finite dimension.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.1 states a commutative diagram whose existence its own proof disclaims; Theorem 5.2 and the uniform Briançon–Skoda/Artin–Rees results rest on this gap.","rationale":"The reader's weakest_assumption identifies the same load-bearing gap: Lemma 5.1's construction does not justify the full commutative diagram used in Theorem 5.2. The manuscript itself flags the problem in the final sentence of the lemma's proof, making this an internal inconsistency rather than a mere disagreement with prevailing expectations. The uniform Briançon–Skoda and Artin–Rees theorems are the most significant new claims, so their dependence on an unproven lemma warrants a CONDITIONAL verdict. I found no other load-bearing concerns: the core derived-category computation in Theorem 2.2 is correct, the splitting arguments for birational derived splinters and reduced blowup-square splinters are natural, and the closure-operation corollaries follow cleanly. The concrete test would settle whether the gap is real, but as written the proof of Section 5 is incomplete.","tokens_in":25486,"tokens_out":17829,"duration_ms":137070,"concrete_test":"Test Lemma 5.1 at the first nontrivial level: set n=1, construct V≤1, X≤1, V●=cosk_1(V≤1), and extend X≤1 to a regular alteration hypercover X● by the proof's inductive procedure. Attempt to define a map V●→X● extending V≤1→X≤1. The proof's final note predicts this may be impossible; concretely exhibiting a non-degenerate 2-simplex of V● whose image in X● cannot be chosen to factor through a regular scheme would confirm the gap. Alternatively, attempt to prove Theorem 5.2 using only the truncated data V≤d+2→X≤d+2; if the H^0-computation cannot be carried out without the full map, the uniform results require a new argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 5.1 asserts, for every n, a commutative diagram with a vertical map nV●→X● between the Zariski hypercover nV● of an alteration nV→X and the alteration hypercover X● with regular terms. The proof constructs nV● := cosk_n(V≤n) and then extends X≤n to X● by weak local uniformization, but its final paragraph admits: 'there might not be a map from nV●→X● fitting the commutative diagram above.' This is an internal inconsistency. Theorem 5.2 invokes this full diagram to claim that the natural map I^{d+k}→RΓ(X●,O_{X●})→L^k(J)⊗^L RΓ(Y●,O_{Y●}) is zero, and later descends from X● to V≤d+2 via X≤d+2. Without a map V●→X● (or a substitute argument using only the truncated diagram), that descent is unsupported. Since Corollaries 5.3 and 5.5 — the uniform Briançon–Skoda and Artin–Rees theorems, i.e., Huneke’s conjectures — depend directly on Theorem 5.2, the principal new uniform results are not established. The rest of the paper (Theorem 2.2 and its consequences for birational derived splinters, Du Bois singularities, and closure operations) appears sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":25782,"tokens_out":28675,"duration_ms":235396,"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":[{"comment":"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.","section":"§5, Theorem 5.2 (also §5, initial outline)"},{"comment":"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.","section":"§5, Lemma 5.1"},{"comment":"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.","section":"§5, proof of Theorem 5.2, descent from X● to V≤d+2"}],"minor_comments":[{"comment":"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'.","section":"Abstract, acknowledgements"},{"comment":"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.","section":"§3, Theorem 3.17, last paragraph"},{"comment":"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.","section":"Abstract vs. §5"}],"recommendation":"major_revision","confidential_remarks":"The core Theorem 2.2 and its non-uniform corollaries are likely correct and already constitute a significant contribution. The Section 5 uniformity claims, however, are not presently justified. The Lemma 5.1 issue is explicitly acknowledged in the manuscript, but the J-vs-I exponent problem in Theorem 5.2 is not acknowledged and is arguably more serious. I would ask the authors to either provide a corrected proof of the uniformity results or remove/rewrite the uniform claims as conditional on additional hypotheses. Given the importance of Huneke's conjectures, a careful revision is warranted rather than outright rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi—\n\nThe short version: Theorem 2.2 is the real deal—a clean derived-category proof of Briançon–Skoda that avoids vanishing and CM assumptions—and the consequences for pseudo-rational rings, birational derived splinters, and Du Bois-type singularities are compelling. The paper would deserve a careful referee on those grounds alone. But the advertised uniform results (Huneke's conjectures) are not proven as written: Lemma 5.1 asserts a commutative diagram with a map V●→X●, and its proof ends by saying such a map may not exist. That is a genuine internal inconsistency, and Theorem 5.2 leans on the lemma.\n\nThat said, the flaw looks fixable, not fatal. In Theorem 5.2, the descent only uses the truncated map V≤d+2→X≤d+2, and the splitting over X● only needs a full alteration hypercover X● with regular terms. Both are present in the construction; the full simplicial compatibility between V● and X● is not actually invoked in the zero-map argument. So the right repair is to weaken Lemma 5.1 to a truncated statement and adjust 5.2 accordingly. The authors should do that before the uniform claims are taken as established.\n\nThe rest of the paper is solid as far as I can see. The closure-operation applications (plus, tight, ep, birational pre-closure) are natural and the citations are appropriate; the self-citations are for context, not load-bearing. I also checked the example in Remark 3.13—the non-Cohen-Macaulay counterexample to a claim in [Li21]—and it looks right. Minor issue: abstract says 'excellent' in the application sentence while the corollaries correctly say 'quasi-excellent'; the proof uses [Lyu25] to reduce, but they should sync the wording.\n\nWho is this for? People working on Briançon–Skoda, closure operations, or uniform bounds. I'd send it to peer review—this is exactly a paper a serious referee should engage with—with the clear request to repair Section 5. I would cite Theorem 2.2 and the splinter corollaries now.\n\nBest,","headline":"The core theorem is elegant and sound; the uniform Section 5 has a repairable but real gap.","tokens_in":26344,"tokens_out":12130,"would_cite":true,"duration_ms":103501,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A15","13B22","13D02","13D45","14B05","13A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["Briançon-Skoda theorem","integral closure","pseudo-rational singularities","birational derived splinters","Buchsbaum-Eisenbud complex","uniform Artin-Rees theorem","uniform Briançon-Skoda theorem","excellent rings"],"falsifier":"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.","tokens_in":25322,"feed_emoji":"🧩","tokens_out":6197,"duration_ms":54229,"temperature":0.7,"texified_at":"2026-08-05T20:33:55.890941+00:00","pith_summary":"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.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":6299,"prompt_tokens":894,"completion_tokens":5405,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":894,"completion_tokens_details":{"reasoning_tokens":4563}},"feed_headline":"Full Briançon-Skoda containment proven for pseudo-rational rings","feed_subtitle":"The same method gives uniform Briançon-Skoda and Artin-Rees bounds for excellent rings.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Briançon-Skoda holds for pseudo-rational singularities","Uniform Briançon-Skoda, Artin-Rees proven for excellent rings","New proof of Briançon-Skoda containment for Du Bois singularities","Characteristic-free Briançon-Skoda via blowup tensor products","Pseudo-rational and Du Bois singularities satisfy Briançon-Skoda"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Briançon-Skoda holds for pseudo-rational singularities","Uniform Briançon-Skoda, Artin-Rees proven for excellent rings","New proof of Briançon-Skoda containment for Du Bois singularities","Characteristic-free Briançon-Skoda via blowup tensor products","Pseudo-rational and Du Bois singularities satisfy Briançon-Skoda"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1365,"prompt_tokens":944,"completion_tokens":421,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":688,"completion_tokens_details":{"reasoning_tokens":326}},"tokens_in":688,"tokens_out":421,"duration_ms":4236,"temperature":1.0,"reasoning_tokens":326,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:07:08.119932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}