{"id":"238e6434-e09d-49f5-8918-50a6315576f2","arxiv_id":"2412.11723","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The constant case of the Grothendieck-Serre conjecture is proved for reductive group schemes over any geometrically regular local algebra over a mixed-characteristic DVR.","lead":"This paper proves that in a hard mixed-characteristic case, a principal bundle that looks trivial over the fraction field was already trivial on the original regular local ring. The proof introduces a new geometric presentation lemma that may be reused for the remaining cases of the Grothendieck-Serre conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's core Proposition 1.7 depends in both residue-field cases on the same-author unpublished theorem [PSt2, Thm. 1.12/Cor. 1.14], which is neither reproduced nor independently verified; if that theorem has hidden hypotheses, Theorem 1.1 is not established.","rationale":"The reader's conditional verdict is appropriate. The main risk is indeed the external dependencies, but I weight [PSt2] more heavily than [P3] because it is used in both cases of Theorem 1.7 and is an unpublished same-author preprint. The Theorem 1.6 moving lemma is a secondary internal gap that would also need to be fixed, but the decisive question is whether [PSt2] holds in the mixed-characteristic local ring setting. Thus the verdict should remain CONDITIONAL pending independent verification of the cited theorem.","tokens_in":15950,"tokens_out":20567,"duration_ms":196588,"concrete_test":"Extract the statements and full proofs of [PSt2, Thm. 1.12 and Cor. 1.14] from arXiv:2305.16627v2, and check the precise hypotheses under which they are proved. Then instantiate the two applications in Theorem 1.7: (i) S = Spec O_{A^{n-1}_V,q(x)} with k(v) infinite, and (ii) S = Spec O_{X,x} with k(v) finite. If either application requires hypotheses not present in the manuscript (for instance, S contains a field, G is quasi-split, or the section over which triviality is known is a rational point with infinite residue field), then the proof of Theorem 1.1 has a genuine gap. A negative result of this check would settle the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Both the infinite and finite residue field cases of Theorem 1.7 use [PSt2, Thm. 1.12 and Cor. 1.14] to conclude that a G-bundle on the relative projective line P^1_S that is trivial at a section is Zariski locally trivial. In the infinite case S = Spec O_{A^{n-1}_V,q(x)}, and in the finite case S = U = Spec O_{X,x}; in both cases S is a mixed-characteristic local ring and G is an arbitrary reductive D-group scheme extended to S. This is a nontrivial relative Gille theorem, and the manuscript cites only arXiv:2305.16627v2 by the same authors without stating its hypotheses or proof. If the theorem actually requires, for example, that S contain a field, that the residue field be infinite, or that G be quasi-split, then the derivation of Zariski local triviality of σ^*E° on P^1_S in the infinite case, and of Ebar_t on P^1_U in the finite case, breaks. Since Theorem 1.7 is the step that converts generic triviality on P°,n_V into Zariski local triviality at each closed point, and Theorem 1.1 relies on this directly, the central claim is only as secure as this external result. The preprint also leaves an unproved moving/transversality assertion in Theorem 1.6 ('we may and will suppose'), but the most load-bearing unverified input is [PSt2].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims Theorem 1.1: for a mixed-characteristic DVR D, a reductive D-group scheme G, and a geometrically regular local D-algebra R containing D, every principal G-bundle over Spec R that is trivial over the fraction field is trivial. This is the 'constant case' of the Grothendieck–Serre conjecture in mixed characteristic. The proof strategy is: reduce by Popescu descent to the case where R is D-smooth, use a theorem of Guo to cut out a divisor of codimension one away from which the bundle is trivial, then prove a DVR version of the Lindel–Ojanguren–Gabber presentation lemma (Theorem 3.5) that produces an elementary distinguished square from a local scheme W of A^n_V. The bundle is then extended over an open subscheme P^{°,n}_V via Theorem 1.6, and Theorem 1.7 is invoked to conclude Zariski local triviality. The paper also develops a substantial geometric presentation lemma for smooth schemes over a DVR in Section 4. The central argument is coherent, but it relies at two load-bearing points on results imported from other papers, [P3] and [PSt2], which are not proved here and whose hypotheses are only partially stated.","tokens_in":16247,"tokens_out":6137,"duration_ms":58707,"significance":"If the external inputs are valid, the paper establishes the constant case of the Grothendieck–Serre conjecture in mixed characteristic, a notable open problem. The presentation lemma of Section 4 is a new tool with independent interest, and the overall reduction strategy is well structured. The paper is not fully self-contained: the finite-residue-field case depends on Theorem 1.8 imported from [P3], and both residue-field cases of Theorem 1.7 depend on the relative Gille theorem from [PSt2]. Those are same-author preprints not reproduced here. The internal proof of Theorem 3.5 is detailed, but it also leaves one verification to the reader in Lemma 4.4 and contains an unproved moving/transversality assertion in the proof of Theorem 1.6. If the missing statements and verifications are supplied or precisely referenced, the result would be a significant advance.","major_comments":[{"comment":"After diagram (4), the proof concludes that the G-bundle σ^*(E°) over P^1_S is Zariski locally trivial solely by citing [PSt2, Corollary 1.14], because its restriction to ∞×S is trivial. This is a nontrivial relative Gille-type theorem over S = Spec O_{A^{n-1}_V,q(x)}, a mixed-characteristic local ring that does not contain a field. The manuscript does not state the theorem, its precise hypotheses, or any proof. If the cited result requires, for example, that the base contain a field or have infinite residue field, or that G be quasi-split, the argument breaks at exactly the point where generic triviality on P^{°,n}_V is converted into Zariski local triviality at a closed point. The authors must include the full statement of [PSt2, Theorem 1.12 and Corollary 1.14] and either prove them or give a complete reference with verified hypotheses.","section":"Proof of Theorem 1.7, infinite residue-field case"},{"comment":"The finite-field case uses two imported results without proof. First, Theorem 1.8 is stated but not proved; it is taken wholesale from [P3, Theorem 1.4]. Second, the conclusion that ar E_t|_{0×U} is trivial is obtained from [PSt2, Theorem 1.12]. The divisor Z produced by Nisnevich's theorem must satisfy the hypotheses of Theorem 1.8, and the manuscript does not verify that the open subscheme X = P^{°,n}_V and the divisor Z meet all the requirements of [P3]. Since this is the only route to local triviality in the finite-field case, the main theorem is not established independently of these external results. The authors should either reproduce the proofs of Theorem 1.8 and the relevant parts of [PSt2] or state their hypotheses in full and confirm they apply to the present divisors.","section":"Proof of Theorem 1.7, finite residue-field case"},{"comment":"After invoking Nisnevich's theorem, the proof says: 'Replacing Y with a divisor Y_ext in P^n_V, containing Y we may and will suppose that x ∉ Y_ext, P^{n-1}_V ⊂ Y_ext and for each irreducible component Z_i of the divisor Z the point x is in Z_i.' This is an unproved moving/transversality assertion. The subsequent decomposition Y_ext ∩ Z = M ∪ Γ_v with M satisfying condition (*) depends on it, and that decomposition is needed to construct the open subset P^{°,n}_V and the bundle E°. A proof or a precise citation for this moving step should be provided.","section":"Proof of Theorem 1.6"},{"comment":"The proof of Lemma 4.4 ends with 'We left to the reader to check that for for these X' and \\tilde{X} the assertions (1) to (7) are true.' This is an omitted verification in a lemma that is load-bearing for Proposition 4.11 and hence for Theorem 3.5. In particular, assertions (5)–(7) concern dimensions of ar X_v − \\tilde X_v and smoothness of the components \\tilde X_{v,j}; these are not immediate formalities. The omitted argument should be supplied.","section":"Lemma 4.4 and its proof"}],"minor_comments":[{"comment":"There are numerous typos and spacing errors, including 'DV R' in the abstract, 'neiborhood', 'inclusin', 'enjoing', and 'subjects the condition' instead of 'satisfies the condition'. These should be corrected.","section":"Throughout"},{"comment":"The notation 'V ⊂ H' for a V-point of H is confusing because V already denotes Spec D. A separate symbol, e.g. s: V → H, would avoid ambiguity.","section":"Proof of Theorem 1.7, infinite residue-field case"},{"comment":"The commutative diagrams are difficult to read because of the arrow labels and line breaks. They should be typeset using a proper commutative-diagram environment.","section":"Diagrams (3) and (4)"},{"comment":"The sentence 'one has dim(\\bar X_{v,i}∩\\bar X_{v,i'}) < n' would be clearer as '≤ n−1', since the strict inequality is already implied by equidimensionality of the closed fibre.","section":"Proof of Lemma 4.4"},{"comment":"The citation [C-T/S, Thm. 6.13] is given for the bijection H^1_et(W,G) → H^1_et(W−S,G). The listed paper is about quadratic fibrations; please confirm that this is the intended reference and state the theorem being cited.","section":"Proof of Theorem 1.6, citation [C-T/S]"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the dependence on the same-author preprints [PSt2] and [P3] for the two decisive steps. I would ask the editors to require the authors to either include full statements and proofs of the imported theorems or replace them by published references with verifiable hypotheses. The missing verification in Lemma 4.4 and the moving step in Theorem 1.6 also need to be addressed. If these points are resolved, the paper would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this is a serious claim to a major open case. If Theorem 1.1 is right, the constant case of Grothendieck–Serre in mixed characteristic is settled, and the DVR presentation lemma is a useful tool. But the proof as written is not self-contained: it leans on two same-author papers, [PSt2] and [P3], for exactly the steps that convert generic triviality into Zariski local triviality. Those steps are load-bearing, so the proof is conditional on results the reader cannot check from this preprint.\n\nWhat is genuinely new: no prior reference states Theorem 1.1 for arbitrary geometrically regular local D-algebras in mixed characteristic. The equal-characteristic and semilocal-Dedekind cases are known, and [GPan3] handles projective smooth schemes over a DVR, but not this setting. The paper also proves Theorem 3.5, a Lindel–Ojanguren–Gabber style presentation lemma over a DVR, with a detailed proof (Section 4) involving finite maps to weighted projective space, Serre vanishing, and miracle flatness. That part is substantive and likely reusable.\n\nThe soft spots, in proportion. The biggest is [PSt2, Cor. 1.14 and Thm. 1.12], used in both residue-field cases of Theorem 1.7. The theorem asserts a Gille-type result: a reductive group scheme over a mixed-characteristic local ring, with triviality at a section of P^1_S, is Zariski locally trivial. That is a nontrivial relative statement. It is only cited, not stated with hypotheses or proof. The stress-test note is correct: if [PSt2] secretly assumes an infinite residue field, or that S contains a field, or that G is quasi-split, the derivation collapses. The finite-field case additionally imports [P3, Thm. 1.4], the 'nice triples' statement, also not reproduced. There is also a 'we may and will suppose' in Theorem 1.6 about extending Y and forcing x to lie in every component of Z; that is an unproved moving/transversality claim. It is probably fillable, but it is not filled here.\n\nThat said, the overall strategy is coherent. Popescu descent reduces to a smooth algebra; [NG] gives a divisor away from which the bundle is trivial; Theorem 1.7 upgrades local triviality via the projective line; and the elementary distinguished square (5) transfers triviality back to X. There is no circularity: Theorem 1.1 is not assumed in the proof of the external results as far as the text shows. The authors are established and the cited companion papers exist, but the burden of supplying their statements and hypotheses should be on this manuscript.\n\nWho should read it: anyone working on torsors, K-theory, or the Grothendieck–Serre conjecture. It deserves a serious referee, but the referee must treat [PSt2] and [P3] as black boxes unless they are verified. My recommendation: send it to peer review with a request that the authors either reproduce the key external results or at minimum state their exact hypotheses and point to where they are proved. As written, I would not accept the proof as complete.","headline":"Major claimed proof, credible but incomplete as written: load-bearing external self-citations must be verified.","tokens_in":16811,"tokens_out":3254,"would_cite":true,"duration_ms":30303,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F20","14L15","14B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The constant case of the Grothendieck–Serre conjecture holds in mixed characteristic: a principal G-bundle over a geometrically regular local algebra is trivial once it is trivial over the fraction field.","keywords":["Grothendieck–Serre conjecture","principal G-bundles","reductive group schemes","mixed characteristic","discrete valuation ring","geometric presentation lemma","elementary distinguished squares","Zariski local triviality"],"falsifier":"Exhibit a mixed-characteristic DVR $D$, a reductive $D$-group scheme $G$, and a geometrically regular local $D$-algebra $R$ carrying a principal $G$-bundle that is trivial over the fraction field of $R$ but not Zariski locally trivial. A smaller counterexample would be an open subscheme $P^{\\circ,n}_V\\subset\\mathbb{P}^n_V$ of the form allowed in Theorem 1.7 with a $G$-bundle trivial at the generic point but not Zariski locally trivial, or a divisor $Z$ through a closed point $x$ of the closed fibre for which the conclusion of Theorem 1.8 fails.","tokens_in":15715,"feed_emoji":"🧮","tokens_out":9265,"duration_ms":80280,"temperature":0.7,"pith_summary":"This paper proves the constant case of the Grothendieck–Serre conjecture in mixed characteristic. Concretely: if $D$ is a discrete valuation ring of mixed characteristic, $G$ is a reductive group scheme over $D$, $R$ is a geometrically regular local $D$-algebra, and a principal $G$-bundle over $\\mathrm{Spec}\\,R$ becomes trivial over the fraction field of $R$, then the bundle is already trivial over $\\mathrm{Spec}\\,R$. The qualifier 'constant' means that $G$ itself is extended from $D$, not chosen freshly over $R$. The result matters because the conjecture was previously known over fields and in several arithmetic settings, while the mixed-characteristic constant case had remained open. As a by-product the paper establishes a DVR-version of the geometric presentation lemma, and that lemma is the technical engine of the proof.","feed_headline":"Constant Grothendieck–Serre case proved in mixed characteristic","feed_subtitle":"A generically trivial principal bundle over a regular local ring is trivial.","key_machinery":"The central mechanism is a geometric presentation lemma over a DVR (Theorem 3.5, the DVR-version of the Lindel–Ojanguren–Gabber lemma). Starting with a smooth affine $V$-scheme $X$, a codimension-one closed subset $Z$ through a closed point $x$ of the closed fibre, and the local scheme $W=\\mathrm{Spec}\\,\\mathcal{O}_{\\mathbb{A}^n_V,y}$ at a closed point $y$ of the affine space fibre, it produces an elementary distinguished square, with $\"\\tau^*(g)=0\" = X'\\cap Z$ as Cartier divisors. This square is what makes the descent work: a bundle trivial off $Z$ can be pulled back, trivialized on the principal open $W_g$, and then reassembled to a bundle over $W$; pulling back along $\\tau$ and using $\\tau(x)=y$ transfers triviality to a Zariski neighborhood of $x$. The construction of the square goes through a finite morphism $\\bar\\pi:\\bar X\\to\\mathbb{P}^{n,w}_V$ to a weighted projective space, built from sections that pass through $x^{(2)}$ and through prescribed finite subsets of the closed fibre.","core_discovery":"The central claim is Theorem 1.1: for a DVR $D$ of mixed characteristic, any reductive $D$-group scheme $G$, any geometrically regular local $D$-algebra $R$, and any principal $G$-bundle $E$ over $\\mathrm{Spec}\\,R$, if $E$ is trivial over the fraction field $K$ of $R$, then $E$ is trivial. The proof splits into two cases according to whether the closed fibre of $X=\\mathrm{Spec}\\,R$ is empty. In the nonempty case the authors choose a closed subset $Z$ of pure codimension one away from which the bundle is trivial, and then reduce the remaining local question to a statement about bundles on open subschemes of projective space over $V=\\mathrm{Spec}\\,D$. That statement, Theorem 1.7, says that a bundle on such an open subscheme that is trivial at the generic point is Zariski locally trivial; it is proved directly when the residue field of $D$ is infinite and via an imported 'nice triples' result when the residue field is finite. The proof is completed by transferring Zariski-local triviality back through an elementary distinguished square to a neighborhood of each closed point of $Z$.","pith_inferences":["A natural next test is whether the same presentation lemma holds when $X$ is only regular rather than smooth over $D$; if it does, the method would likely reach a broader class of bases than DVRs.","The paper's dependence on two imported statements is itself a roadmap: anyone proving those statements for a more general class of schemes would automatically extend the main theorem.","The weighted-projective-space construction used to build the presentation square may be reusable for other local-to-global problems, since it converts a divisorial neighborhood problem into a finite flat morphism problem."],"forward_implications":["For every constant reductive group scheme over a mixed-characteristic DVR, rationally trivial principal bundles over geometrically regular local algebras are Zariski locally trivial.","The DVR-version of the geometric presentation lemma becomes available as a tool for other torsor and motivic questions in mixed characteristic.","The extension theorem (Theorem 1.6) and the generic-triviality theorem (Theorem 1.7) together give a general way to pass from affine space to projective space when studying generically trivial bundles.","Since the proof covers both infinite and finite residue fields, the conjecture is settled uniformly whenever the imported 'nice triples' and projective-line statements hold."],"supporting_citations":[{"why":"Supplies the Grothendieck–Serre result over regular local rings containing an infinite field, used for the empty-closed-fibre case and for points away from the closed fibre.","marker":"[FP, Theorem 1.1]"},{"why":"Produces the pure-codimension-one closed subset $Z$ away from which the bundle is trivial, the starting point of the local argument.","marker":"[NG, Theorem 1]"},{"why":"Provides Zariski local triviality for bundles on the relative projective line over a local scheme, used in both the infinite and finite residue field cases.","marker":"[PSt2, Theorem 1.12 and Corollary 1.14]"},{"why":"Supplies the 'nice triples' geometric statement (Theorem 1.8) that is the load-bearing input in the finite residue field case.","marker":"[P3, Theorem 1.4]"},{"why":"Gives the bijection $H^1_{\\mathrm{et}}(W,G)\\to H^1_{\\mathrm{et}}(W-S,G)$ used to extend bundles past codimension-two loci.","marker":"[C-T/S, Thm. 6.13]"},{"why":"Popescu's desingularization theorem lets the proof reduce from a geometrically regular local $D$-algebra to a smooth integral domain.","marker":"[Po]"},{"why":"Defines elementary distinguished squares, the patching mechanism used throughout the descent and presentation arguments.","marker":"[MV, Defn.3.1.3]"},{"why":"Provides the divisor of triviality for a generically trivial bundle, used to set up the reduction to the projective-space statements.","marker":"[N]"}],"fun_headline_variants":["Constant Grothendieck-Serre conjecture proven in mixed characteristic","Mixed char: constant Grothendieck-Serre conjecture true","Constant Grothendieck-Serre case settled in mixed characteristic","Mixed characteristic proof for constant Grothendieck-Serre","Grothendieck-Serre constant case proved over mixed characteristic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The finite-residue-field half of the proof depends on two results taken from other preprints: a geometric 'nice triples' statement (Theorem 1.8) that supplies a monic-polynomial slice through a divisor, and a statement that bundles on the projective line over a local scheme are Zariski locally trivial. If either of those imported results is wrong, or if the nice-triples statement does not apply to the divisor produced by the standard triviality theorem, the proof of the main theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Constant Grothendieck-Serre conjecture proven in mixed characteristic","Mixed char: constant Grothendieck-Serre conjecture true","Constant Grothendieck-Serre case settled in mixed characteristic","Mixed characteristic proof for constant Grothendieck-Serre","Grothendieck-Serre constant case proved over mixed characteristic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001035,"raw_usage":{"total_tokens":4306,"prompt_tokens":840,"completion_tokens":3466,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":3381}},"tokens_in":456,"tokens_out":3466,"duration_ms":25730,"temperature":1.0,"reasoning_tokens":3381,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:39:47.008770+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a mixed-characteristic DVR $D$, a reductive $D$-group scheme $G$, and a geometrically regular local $D$-algebra $R$ carrying a principal $G$-bundle that is trivial over the fraction field of $R$ but not Zariski locally trivial. A smaller counterexample would be an open subscheme $P^{\\circ,n}_V\\subset\\mathbb{P}^n_V$ of the form allowed in Theorem 1.7 with a $G$-bundle trivial at the generic point but not Zariski locally trivial, or a divisor $Z$ through a closed point $x$ of the closed fibre for which the conclusion of Theorem 1.8 fails.","supporting_citations":[],"review_version":1}