{"id":"952864eb-f7f1-41fd-9e50-254d47408bd8","arxiv_id":"2501.01567","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"K2-regularity forces normality in all dimensions, and for affine local complete intersections K(p+1)-regularity forces regularity in codimension 2p.","lead":"A new proof shows that algebras whose second algebraic K-group has a strong vanishing property must be normal, meaning all their local rings are integrally closed. The paper also sharpens the known K-theoretic test for smoothness of local complete intersections, linking algebraic K-theory to du Bois singularities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.6's 'by inspection' step drops condition (2.5a) from Lemma 2.5; the proof does not show that p-du Bois alone yields the cdh-cohomology comparison needed to conclude Ω^{d+1}=0.","rationale":"I read the full text in good faith. Theorem 0.1 rests on the Milnor-square arguments in Section 1; the use of the conductor, excision, and the Dennis–Stein symbol appears internally consistent, and I did not find a concrete flaw there. The reader's stated dependency is partially inaccurate: Theorem 0.3 is proved directly from Proposition 2.7 and Mustata–Popa's Theorem 3.1, not from Theorem 2.6, so a failure of Theorem 2.6 would not by itself destroy Theorem 0.3 or Corollary 3.6. Nevertheless, the genuine weak point is exactly the 'by inspection' passage in the proof of Theorem 2.6. Lemma 2.5 characterizes p-cdh descent by three simultaneous conditions; the p-du Bois hypothesis supplies only the two Hodge-theoretic conditions (2.6a,b), while the paper gives no detailed derivation that these suffice to rerun [1, Theorem 3.1]. In particular, condition (2.5a) on Hochschild homology is not evidently implied by p-du Bois, and the proof sketch does not show where it is or is not used. Because this is a claimed theorem in the paper and supports the Section 2 reinterpretation of Vorst's conjecture, the conditional verdict is appropriate. My read does not change the reader's verdict, so I recommend UNCHANGED.","tokens_in":9119,"tokens_out":31393,"duration_ms":308359,"concrete_test":"Independently write out the proof of [1, Theorem 3.1] for a local d-dimensional ring A using only hypotheses (2.6a,b), tracking every use of condition (2.5a). In particular, construct the morphism H*(Ω^{≤d}_{A/k}, d) → H*_cdh(X, Ω^{≤d}_{/k}) and verify explicitly that it is an isomorphism without invoking the Hochschild homology vanishing (2.5a). If the verification requires (2.5a), Theorem 2.6 fails as stated. As a complementary computational check, test the conclusion on a known non-lci p-du Bois singularity, such as a 1-du Bois normal surface singularity, by computing Ω^{d+1}_{A/k} in Macaulay2; a nonzero Ω^{d+1} would refute the theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing concern is in the proof of Theorem 2.6 (Section 2, pages 7-8). The theorem asserts that p-du Bois singularities imply regularity in codimension p. The proof reduces to a d-dimensional local ring A with d ≤ p, states that the p-du Bois condition is 'equivalent to' conditions (2.6a,b), and then says 'by inspection' the proof of [1, Theorem 3.1] goes through with only these hypotheses, yielding H*(Ω^{≤d}_{A/k}, d) ≅ H*_cdh(X, Ω^{≤d}_{/k}) ≅ H*_cdh(X, Ω^•_{/k}) and hence Ω^{d+1}_{A/k} = 0. The problem is that Lemma 2.5 lists three conditions for p-cdh descent; condition (2.5a), the vanishing of Hochschild homology HH^q_m(R/k) for 0 ≤ q < m ≤ p, is not a consequence of p-du Bois. The paper gives no argument that the comparison of de Rham cohomology with cdh-cohomology, and the final implication to Ω^{d+1}=0, can be obtained from the Hodge-theoretic du Bois isomorphisms alone. If any step in [1, Theorem 3.1] uses (2.5a) to identify these cohomology groups, then Theorem 2.6 is unsupported. This does not directly invalidate Theorem 0.3, whose proof uses Proposition 2.7 plus the Mustata–Popa Theorem 3.1, but it leaves the paper's Section 2 claim and the new proof of Vorst's conjecture conditional on an unverified assertion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper has two main parts. In Section 1, using a conductor square, relative K-theory exact sequences, and a Dennis--Stein symbol argument after strict henselization, the authors prove that K2-regularity implies normality for noetherian rings containing Q with finite normalization (Theorem 0.1/1.9), and also treat the two-dimensional perfect-residue-field case. In Sections 2 and 3, the paper connects K-regularity to higher du Bois singularities. Proposition 2.7 extracts from the published [1] that K_{p+1}-regularity implies p-du Bois; Theorem 2.6 claims conversely that p-du Bois singularities are regular in codimension p; and Theorem 3.3 combines Proposition 2.7 with the Mustata--Popa theorem to prove that K_{p+1}-regular local complete intersections are regular in codimension 2p. Theorem 3.5, credited to Wanchun Shen, sharpens this using minimal exponents.","tokens_in":9426,"tokens_out":24193,"duration_ms":241659,"significance":"If the proofs are completed, the paper would contain two notable results: a dimension-free normality criterion for K2-regular rings, and a substantial improvement over Vorst's bound for local complete intersections. The Section 1 induction is largely independent of the authors' earlier work and appears sound in outline; the Section 3 lci argument is short and transparent because it separates the K-theoretic input (Proposition 2.7) from the Hodge-theoretic input (Mustata--Popa). The paper is concise and does not overclaim in its main statements. However, the Section 2 result Theorem 2.6, which is presented as a general structural consequence of du Bois conditions, is proved only by an assertion that is not verified in the text. That gap does not by itself invalidate Theorem 0.1 or Theorem 0.3, but it is load-bearing for Theorem 2.8 and for the categorical statement in Question 3.2, and it must be fixed or removed before the paper can be accepted.","major_comments":[{"comment":"The proof is incomplete in a load-bearing way. The reduction of the p-du Bois hypothesis to conditions (2.6a)–(2.6b) is not literally Definition 2.3, which requires isomorphisms for all q ≤ p rather than only q ≤ d; the case d < q ≤ p is dropped without explanation. More seriously, the sentence \"By inspection, the proof of [1, Theorem 3.1] only needs the weaker hypotheses given here\" is the actual content of the theorem, and the text does not check it. Lemma 2.5 lists three conditions characterizing p-cdh descent; condition (2.5a), the vanishing of the Hochschild homology groups HH^q_m(R/k) for 0 ≤ q < m ≤ p, is not shown to follow from p-du Bois, and the displayed comparison H*(Ω^{≤d}_{A/k}, d) ≅ H*_cdh(X, Ω^{≤d}_{/k}) ≅ H*_cdh(X, Ω^•_{/k) is precisely the kind of statement for which (2.5a) is used in [1]. Since the final conclusion Ω^{d+1}_{A/k} = 0 is obtained from these isomorphisms, Theorem 2.6 is unsupported as written. Consequently Theorem 2.8 and the assertion in Question 3.2 that a 1-du Bois algebra is regular in codimension 1 inherit the gap.","section":"§2, Theorem 2.6 (pp. 7–8)"},{"comment":"The construction of the elements f and g after strict henselization is not justified. The proof needs f,g in the maximal ideal with disjoint supports among the factors of B', so that fg = 0, and with linearly independent classes modulo m^2. Such elements depend on a branch decomposition of the strict local ring A'; if A' is a domain then B' is a domain and s > 1 is already impossible, while if A' is not a domain, the existence of branch-separating functions in A' and their linear independence modulo m^2 must be proved. Because the Dennis--Stein symbol and the nonvanishing of its Dennis trace depend on these properties, the contradiction proving A = B, and hence Theorem 1.9, rests on an unstated branch-separation lemma.","section":"§1, Proposition 1.8 (pp. 4–5)"}],"minor_comments":[{"comment":"The displayed isomorphism \"N U(A/c) ∼= -→ N U(B/c)\" contains a stray arrow and should be cleaned up.","section":"§1, Corollary 1.4"},{"comment":"The notation Ω^p_{/k}|X in the definition of Ω^p_{X/k} is confusing; please specify that Ω^p_{/k} is first regarded as a sheaf on the cdh site and then restricted to X.","section":"§2, Definition 2.1"},{"comment":"The theorem is credited to Wanchun Shen as a personal communication; if a published source is available, it should be cited here, and otherwise the provenance should be stated precisely so that the reader can verify the hypotheses.","section":"§3, Theorem 3.5"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the Section 1 proof of Theorem 0.1 is the genuinely new part and is mostly sound; the gap in Proposition 1.8 appears repairable with a short argument about minimal primes of the strict henselization. The Section 2 gap is more serious because Theorem 2.6 is advertised as a general structural result. If the authors cannot supply the missing proof, they should remove Theorem 2.6 and its dependent statements from the claims; Theorem 0.3 does not rely on Theorem 2.6. The reliance on [1] is not circular for Theorem 0.1 and is acceptable for Proposition 2.7 because [1] is a published, refereed result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nTwo results here are solid and new. Theorem 0.1 generalizes Vorst's curve theorem: K2-regularity forces normality in arbitrary dimension for noetherian rings containing Q with finite normalization. The Milnor square argument with the conductor is coherent; the use of the Dennis-Stein symbol to detect non-smoothness is clever and, as far as I can tell, sound. Theorem 0.3 improves Vorst's bound for affine local complete intersections: K(p+1)-regularity gives regularity in codimension 2p, roughly halving the old threshold. This is a clean combination of Proposition 2.7 (extracted from the authors' 2008 JAMS paper) and Mustata-Popa's Hodge-theoretic theorem.\n\nThe soft spot is Theorem 2.6, which states that p-du Bois singularities imply regularity in codimension p. The proof says 'by inspection' that the proof of [1, Theorem 3.1] only needs the two conditions (2.6a) and (2.6b), and drops condition (2.5a) from Lemma 2.5, the vanishing of intermediate Hochschild homology. I share the stress-test concern: p-du Bois is a statement about de Rham/du Bois complexes, and I see no reason it should kill those Hochschild groups. Without (2.5a), the comparison H*(Omega^{≤d}_{A/k}, d) ≅ H*_cdh(X, Omega^{≤d}) is not justified, so the conclusion Ω^{d+1}=0 hangs in the air. Since Theorem 2.6 supports Theorem 2.8 (a new proof of Vorst's conjecture), that part of the paper is not yet established. Importantly, Theorem 2.6 is not needed for Theorem 0.3, which uses Proposition 2.7 and Theorem 3.1 directly. So the paper's main new contributions survive, but the advertised stronger statement about du Bois singularities should be either proved with the missing hypothesis or downgraded.\n\nThe rest is in good shape. Section 1 tracks all hypotheses (m-primary conductor, separability). The paper is explicit about what is from [1] and what is new, and the citations are appropriate. I found no invented entities and no data fitting.\n\nWho should read it: algebraic K-theorists and anyone interested in singularity criteria. Theorem 0.1 is likely to become a standard reference; Theorem 0.3 is a meaningful sharpening for lci singularities.\n\nRecommendation: yes, send it to a serious referee. Ask the referee to focus on Theorem 2.6 and whether the 'by inspection' step can be made rigorous, and if not, whether the authors can add (2.5a) as an explicit hypothesis or weaken the claim. The rest is publishable as is.","headline":"Theorem 0.1 and 0.3 are solid and worth knowing; Theorem 2.6's 'by inspection' gap is real and should be fixed before the du Bois-to-regularity claim is taken as established.","tokens_in":10049,"tokens_out":3742,"would_cite":true,"duration_ms":33258,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19D35","14F20","19E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"K2-regularity, a vanishing condition on polynomial-extended algebraic K-theory, forces normality in characteristic zero, and for local complete intersections it forces regularity from about half of Vorst's bound.","keywords":["algebraic K-theory","K-regularity","normality","du Bois singularities","local complete intersections","cdh descent","Hochschild homology","minimal exponent"],"falsifier":"Find a 3-dimensional local complete intersection singularity over C that is K2-regular but has minimal embedding codimension 2; Corollary 3.6 predicts r ≤ 1, so such an example would refute Theorem 0.3 and the Shen sharpening.","tokens_in":8861,"feed_emoji":"🧮","tokens_out":7388,"duration_ms":64657,"temperature":0.7,"pith_summary":"This paper tries to show that K-theoretic regularity—a condition on the vanishing of certain Nil K-groups—carries strong geometric information about singularities. Its first theorem states that in characteristic zero, any noetherian ring with finite normalization that is K2-regular must be normal, a conclusion previously known only in low dimensions. The second main theorem improves Vorst's classical bound for affine local complete intersections: K(p+1)-regularity forces regularity in codimension 2p, so a d-dimensional local complete intersection that is K(q)-regular for q ≥ d/2 + 1 is actually regular. If these results stand, then a single low-degree K-theoretic vanishing condition can certify normality or even smoothness without dimension restrictions.","feed_headline":"K2-regularity forces normality in characteristic zero","feed_subtitle":"A single K-theoretic condition controls normality in all dimensions and sharpens Vorst's bound.","key_machinery":"The argument runs on three machines. The normality half uses the conductor square of a finite birational extension A ⊆ B: applying relative K-theory and excision to the Milnor square reduces K2-regularity of A to a statement about the residue-field extension, where Lemma 1.5 identifies the obstruction with the relative group $K_1(A[t], B[t], \\mathfrak{m}[t]) \\cong \\mathfrak{m}/\\mathfrak{m}^2 \\otimes_L \\Omega_{L/K} \\otimes_L L[t]$; a nonzero K2 element is produced by a Dennis–Stein symbol whose Dennis trace to $\\Omega^2_{A'[t]}$ is a nonzero 2-form. The regularity half uses p-du Bois complexes $\\underline{\\Omega}^p_X$, defined via cdh-descent from the sheaf of Kähler differentials, and the theorem that K(p+1)-regularity implies p-du Bois singularities through p-cdh descent for Hochschild homology. For local complete intersections, the bound is carried by the minimal exponent $\\widetilde{\\alpha}(S,X)$: combining the upper bound $\\widetilde{\\alpha} \\le n/2$ from the local complete intersection condition and the lower bound $\\widetilde{\\alpha} \\ge p + r$ from p-du Bois behaviour forces regularity in codimension $2p + r - 1$.","core_discovery":"On the paper's own terms, the central discovery is that the vanishing of polynomial-extended K-groups is a sharper singularity detector than previously recognized. Theorem 0.1 asserts that K2-regularity forces normality for noetherian rings containing Q whose normalization is finite, and the proof works by localizing at a prime, using the conductor square to reduce to the case where the conductor is the maximal ideal, and then showing that a nontrivial finite birational extension would produce a nonzero Dennis–Stein symbol, hence a nonzero element of N K2. Theorem 0.3 asserts that for affine local complete intersections over a characteristic-zero field, K(p+1)-regularity implies regularity in codimension 2p, so that K(q)-regularity with q at least d/2 + 1 implies regularity; the proof passes through p-du Bois singularities and the Hodge-theoretic theorem that p-du Bois local complete intersection singularities are regular in codimension 2p. The paper also records a sharpening, communicated by Wanchun Shen, showing that a p-du Bois local complete intersection singularity of embedding codimension r is regular in codimension 2p + r − 1, and it recovers a new proof of Vorst's conjecture as a special case.","pith_inferences":["If Theorem 0.1 is right, K2-regularity is a purely K-theoretic way to detect normality without fixing the dimension; one testable consequence is that K2-regularity should fail for every non-normal isolated singularity in characteristic zero, which could be checked by computing N K2 of its local ring directly.","The separation between the normality theorem (no local complete intersection assumption) and the smoothness theorem (local complete intersection assumption) suggests that the true obstruction to smoothness under K2-regularity is torsion-freeness of differentials; non-local-complete-intersection K2-regular surfaces would be a natural place to look for a counterexample to regularity.","The minimal-exponent formulation in Theorem 3.5 gives an explicit inequality r ≤ d − 2p linking embedding codimension to K-regularity degree; this could be tested computationally on explicit local complete intersection singularities, since the minimal exponent is accessible from a V-filtration.","In positive characteristic, the proof of normality requires separability of residue-field extensions (Lemma 1.5), suggesting K2-regularity alone may fail to imply normality over imperfect fields; constructing an inseparable K2-regular local ring would settle the sharpness of the Q ⊆ A hypothesis."],"forward_implications":["In characteristic zero, K2-regularity becomes a dimension-free certificate of normality for affine algebras with finite normalization, extending Vorst's curve-level result to all dimensions.","For affine local complete intersections of dimension d, Vorst's bound K_{d+1}-regularity can be replaced by K_q-regularity with q ≥ d/2 + 1; the same conclusion—regularity—follows from roughly half as much K-theoretic input.","A 3-dimensional K2-regular local complete intersection singularity must be a hypersurface, since its minimal embedding codimension r must satisfy r ≤ d − 2p = 1.","Every K_{p+1}-regular affine scheme is regular in codimension p, giving a new proof of Vorst's conjecture through p-du Bois singularities rather than through the full descent conditions."],"supporting_citations":[{"why":"Supplies the Vorst-conjecture proof and the cdh-descent/Hochschild homology machinery that Theorem 2.6 reruns under weakened hypotheses.","marker":"[1]"},{"why":"Supplies the K-theoretic calculations on N K groups and cdh-fibrant Hochschild homology behind the local complete intersection bound.","marker":"[2]"},{"why":"Establishes the cdh/Zariski comparison that defines the du Bois complexes used throughout Section 2.","marker":"[3]"},{"why":"Provides the minimal-exponent bounds for local complete intersections used in the sharpening Theorem 3.5.","marker":"[6]"},{"why":"Defines Dennis–Stein symbols, used to produce a nonzero K2 element obstructing normality in Proposition 1.8.","marker":"[8]"},{"why":"Introduces the filtered de Rham/du Bois complexes that define p-du Bois singularities.","marker":"[9]"},{"why":"Computes the relative K1 group whose vanishing characterizes separability in Lemma 1.5.","marker":"[14]"},{"why":"Proves the Hodge-theoretic theorem that p-du Bois local complete intersection singularities are regular in codimension 2p, the engine of Theorem 3.3.","marker":"[21]"},{"why":"Vorst's localization theorem and K-regularity bounds, the starting point for both the normality argument and the improved local complete intersection bound.","marker":"[26]"}],"fun_headline_variants":["K2-regularity forces normality over Q","K2-regularity implies normality in char zero","K2-regularity sharpens Vorst's bound","Non-normal rings fail K2-regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the p-du Bois condition provides exactly the two vanishing conditions (2.6a) and (2.6b), and that these suffice to carry through the earlier proof and force the next exterior power of differentials to vanish; the paper asserts this bridge by inspection rather than proving it.","fun_headline_variants_meta":{"raw":{"variants":["K2-regularity forces normality over Q","K2-regularity implies normality in char zero","K2-regularity sharpens Vorst's bound","Non-normal rings fail K2-regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000597,"raw_usage":{"total_tokens":2746,"prompt_tokens":848,"completion_tokens":1898,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":1836}},"tokens_in":464,"tokens_out":1898,"duration_ms":13933,"temperature":1.0,"reasoning_tokens":1836,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:26:43.536127+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a 3-dimensional local complete intersection singularity over C that is K2-regular but has minimal embedding codimension 2; Corollary 3.6 predicts r ≤ 1, so such an example would refute Theorem 0.3 and the Shen sharpening.","supporting_citations":[{"cited_title":"Corti˜ nas, C","cited_arxiv_id":null,"evidence_quote":"Supplies the Vorst-conjecture proof and the cdh-descent/Hochschild homology machinery that Theorem 2.6 reruns under weakened hypotheses."},{"cited_title":"Corti˜ nas, C","cited_arxiv_id":null,"evidence_quote":"Supplies the K-theoretic calculations on N K groups and cdh-fibrant Hochschild homology behind the local complete intersection bound."},{"cited_title":"Corti˜ nas, C","cited_arxiv_id":null,"evidence_quote":"Establishes the cdh/Zariski comparison that defines the du Bois complexes used throughout Section 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the minimal-exponent bounds for local complete intersections used in the sharpening Theorem 3.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines Dennis–Stein symbols, used to produce a nonzero K2 element obstructing normality in Proposition 1.8."},{"cited_title":"du Bois, Complexe de de Rham filtr´ e d’une vari´ et´ e singuli` ere, Bull","cited_arxiv_id":null,"evidence_quote":"Introduces the filtered de Rham/du Bois complexes that define p-du Bois singularities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes the relative K1 group whose vanishing characterizes separability in Lemma 1.5."},{"cited_title":"Pi 10 (2022), Paper No","cited_arxiv_id":null,"evidence_quote":"Proves the Hodge-theoretic theorem that p-du Bois local complete intersection singularities are regular in codimension 2p, the engine of Theorem 3.3."},{"cited_title":"Ann., 244:33–54, 1979","cited_arxiv_id":null,"evidence_quote":"Vorst's localization theorem and K-regularity bounds, the starting point for both the normality argument and the improved local complete intersection bound."}],"review_version":1}