{"id":"09c36062-2f40-42b2-a00d-d4613ae8d778","arxiv_id":"2506.10637","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Q-schemes and Dedekind schemes, the motivic t-structure exists on 1-motives with integral coefficients, with heart the abelian category of Deligne 1-motives with torsion.","lead":"This mathematics paper constructs a key organizing structure for 1-motives, geometric objects built from abelian varieties and tori, with integer coefficients over Q-schemes and Dedekind schemes. It proves the resulting heart is the abelian category of Deligne 1-motives with torsion, a step toward Grothendieck's unifying theory of cohomology.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 5.4's extension from regular bases to arbitrary Q-schemes rests on an unverified reduction to [Rui22, Theorem 4.2.7] for integral 1-motives; if that reduction fails, the main theorem is only proved in the regular (Dedekind) case.","rationale":"The reader's conditional verdict is appropriate, and the weakest assumption is the same one I would identify. The main theorem's extra content over Theorem 5.1 is precisely the passage from regular S to arbitrary Q-schemes, and the proof of Corollary 5.4 delegates that passage to [Rui22, Theorem 4.2.7], a theorem about integral Artin 0-motives. Since the surrounding machinery for 1-motives with integral coefficients is new and not identical to the 0-motive case, this delegation is not a formal proof. The gap is real but plausibly fillable: the paper contains many of the required tools, including Theorem 2.8, Proposition 3.3, Corollary 4.13, and Theorem 5.1. Therefore the appropriate verdict remains CONDITIONAL rather than REJECT: the central claim is credible but currently rests on an unverified reduction. Because my concern does not move the reader's verdict, I set verdict_should_be to UNCHANGED.","tokens_in":25386,"tokens_out":9285,"duration_ms":117335,"concrete_test":"Write out the proof of Corollary 5.4 for a non-regular reduced finite-type Q-scheme S, for instance a nodal affine curve, following [Rui22, Theorem 4.2.7] step by step. For each 0-motive lemma used there, verify its 1-motive analogue using only results proved in this paper. The three steps that must be checked are: (i) the class DG_S of Definition 4.1 satisfies the hypotheses needed for [Rui22, Proposition 1.1.10] to apply to Ind-1-motives; (ii) for a closed immersion i and open immersion j, if τ≤0(j*M) and τ≤0(i*M) are 1-motives then τ≤0(M) is a 1-motive; (iii) over a regular base, every p-geometric object is generated by smooth 1-motives. If any of these steps cannot be filled except by citing the 0-motive result, the reduction is not a routine citation and Corollary 5.4 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem, Corollary 5.4, asserts the motivic t-structure restricts to DM1(S,Λ) for every Q-scheme and every Dedekind scheme. Theorem 5.1, the only place where the restriction is actually proved, requires S to be regular and to satisfy Hypothesis 2.3. Dedekind schemes are regular under the usual convention, so the genuinely new content of Corollary 5.4 is the case of singular Q-schemes. Its proof says only 'exactly as in the proof of [Rui22, Theorem 4.2.7], we can reduce...' and then applies Theorem 5.1. None of that reduction is written out for 1-motives with integral coefficients. The analogous reduction for 0-motives in [Rui22] relies on structure special to Artin motives: smooth 0-motives are exactly lisse sheaves, the ordinary t-structure on D_lisse is explicit, and localization triangles are controlled through p-geometric objects. For integral 1-motives, none of these ingredients is established over a singular base in this paper. In particular, it is not shown that every 1-motive is glued from smooth 1-motives over regular strata, nor that τ≤0 preserves p-geometric Ind-1-motives for the 1-motive generator class, nor that the non-excellent case is covered, since [Rui22] assumes excellent schemes with resolution by alterations. If this reduction is not valid, Theorem 5.1 yields only the regular case, and Corollary 5.4 is unproved for singular Q-schemes. This is exactly the fragile step identified by the reader.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Deligne 1-motives with torsion over a noetherian base scheme as a Gabriel-Zisman localization of effective 1-motives, and proves that this localization is a 1-category. Over a connected normal scheme it establishes full faithfulness of the generic-point restriction functor for coefficients in a flat Z′-algebra, together with a good-reduction criterion in terms of ℓ-adic Tate modules, from which it deduces that MD1(S,Λ) is abelian when Λ is a localization of Z′. It then defines a motivic t-structure on the category of Ind-1-motives and proves compatibility with pullbacks, localizations, and ℓ-adic realizations, and shows that the functor ΦS from Deligne 1-motives to étale motives lands in the heart. The main theorem (Corollary 5.4) asserts that, for S a Q-scheme or a Dedekind scheme and Λ a localization of Z, this t-structure restricts to DM1(S,Λ) and is compatible with pullback and ℓ-adic realization.","tokens_in":25733,"tokens_out":14017,"duration_ms":136107,"significance":"If the main theorem is correct, it is a significant advance: it provides an integral motivic t-structure on relative 1-motives over bases that need not be regular or of finite type over a field, extending the field-coefficient results of [BVK16], the rational-coefficient results of [Leh19b, Leh19a, Vai19], and the author's earlier work on integral Artin motives. The paper also contains independent contributions: the proof that Deligne 1-motives with torsion form an abelian category after inverting residue characteristic exponents, and a good-reduction criterion generalizing work of Haas. Many local steps are proved in detail (e.g., Theorem 1.5, Theorem 2.8, Proposition 4.12, Lemma 5.3), and the paper is carefully structured. The main caveat is that the passage from the regular case to arbitrary Q-schemes in Corollary 5.4 is not fully proved; this is the load-bearing step that the rest of the report addresses.","major_comments":[{"comment":"The proof of the main theorem relies on the assertion 'exactly as in the proof of [Rui22, Theorem 4.2.7], we can reduce to the case when ...' without carrying out the reduction. This is load-bearing: Theorem 5.1, the only place where the motivic t-structure is actually shown to restrict to smooth 1-motives, assumes S is regular, and the entire novelty of Corollary 5.4 beyond the Dedekind case is the treatment of singular Q-schemes. The analogous reduction for 0-motives in [Rui22] uses structure that is not established here for integral 1-motives: smooth 0-motives are exactly lisse sheaves, the ordinary t-structure on D_lisse is explicit, and p-geometric objects are controlled through those identifications. For integral 1-motives over a singular base, the manuscript does not prove that every 1-motive is glued from smooth 1-motives over regular strata, nor that τ≤0 preserves p-geometric Ind-1-motives for the 1-motive generator class, nor that the non-excellent case is covered (the statement allows arbitrary noetherian finite-dimensional Q-schemes, while [Rui22] assumes excellent schemes allowing resolution by alterations). Without a written proof of this reduction, Corollary 5.4 is not established for singular Q-schemes.","section":"Section 5, proof of Corollary 5.4"},{"comment":"The statement of Corollary 4.16 claims that ΦS lands in the heart of the motivic t-structure for an arbitrary scheme S and arbitrary flat Z-algebra Λ, but its proof invokes Corollary 4.13, which assumes S is excellent and allows resolution of singularities by alterations. No justification is given for why these hypotheses can be dropped, or for why they hold in the applications of Corollary 4.16 (e.g., in the proof of Corollary 5.4, where the reduced scheme is not explicitly shown to be regular). If Corollary 4.16 is needed for the main theorem, its hypotheses and proof must be aligned; otherwise the statement should be restricted to the cases actually used.","section":"Section 4, Corollary 4.16"},{"comment":"Remark 4.14 asserts that the method of [Leh19a] works with Λ⊗Q coefficients for any flat Z-algebra Λ 'by direct inspection' and invites the reader to check. This is not a proof, and it is particularly problematic because several statements in the paper (e.g., Theorem 3.5 and Corollary 4.16) are formulated for arbitrary flat Z-algebras. Since the main theorem only needs Λ to be a localization of Z, the paper would be cleaner if these statements were restricted accordingly, or if a proof of the asserted extension were supplied.","section":"Section 4, Remark 4.14"}],"minor_comments":[{"comment":"There are a few typographical glitches, such as 'on1-motives' in the abstract and 'finial section' in the introduction; these should be corrected.","section":"Abstract and Introduction"},{"comment":"The parenthetical 'for any ℓ that is invertible on S (and thus any non-invertible prime in Z′)' appears to have the primes reversed; it should say 'any prime ℓ≠p' or 'any prime invertible in Z′'.","section":"Section 2, proof of Proposition 2.10"},{"comment":"The reference '[Ray70, Collaire IX.1.4]' should presumably read 'Corollaire IX.1.4'.","section":"Section 2, proof of Theorem 2.8"},{"comment":"The claim that the assumptions on cohomological dimension in [Ayo14, Proposition 11.1] can be removed by the rigidity theorem is stated without proof; please provide a reference for this removal or a short argument.","section":"Section 3, proof of Lemma 3.7"},{"comment":"In the proof that the kernel of a map of Deligne 1-motives is again a Deligne 1-motive, the exactness of the functor between the abelian categories is asserted; a brief explanation of why the family i_x^* is exact and conservative (using Lemma 4.15 and Proposition 4.12) would improve readability.","section":"Section 5, proof of Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own preprints ([Rui22], [Rui25], [RT24b]), several of which are unpublished. This is not inherently a problem, but the main theorem's proof delegates its key reduction to [Rui22] without enough detail for a reader to verify it. Given the breadth of the main statement (arbitrary noetherian Q-schemes), the editor may wish to ask the author to make the reduction explicit or to narrow the statement to cases where the proof is complete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper does something real: it extends the motivic t-structure to 1-motives with integral coefficients over Q-schemes and Dedekind schemes, and identifies the heart with Deligne 1-motives with torsion once you invert residue characteristic exponents. The rational-coefficient case was known (Lehalleur, Haas, Vaish); the integral case over base schemes is new. The paper also proves that Deligne 1-motives with torsion form an abelian category over normal schemes under the good-reduction hypothesis, and gives a torsion-friendly criterion for good reduction in terms of Tate modules. Those results are useful beyond the main theorem.\n\nThe arguments are mostly serious and carefully built. The t-structure on Ind-1-motives is defined by generators, compared with the field case, and shown to interact well with torsion and pullback. Section 5 proves the regular case in Theorem 5.1 with real work, and the deduction of the full theorem is plausible.\n\nThe soft spot is exactly the step the stress-test flags. Corollary 5.4 goes from regular bases to arbitrary Q-schemes by saying 'exactly as in the proof of [Rui22, Theorem 4.2.7]', without writing out the reduction for 1-motives with integral coefficients. The paper does not show that singular schemes can be glued from regular strata preserving p-geometricity for 1-motives, that every 1-motive is glued from smooth 1-motives over regular strata, or that the non-excellent case is covered. If that reduction is not valid, the main theorem only covers regular bases. This is a load-bearing gap, though not evidence of a false theorem. It needs to be spelled out before the result is fully accepted.\n\nThere is also a smaller handwave: Remark 4.14 asks the reader to check by inspection that Lehalleur's rational-coefficient machinery works with integral coefficients after inverting the localization. That is probably fine, but it should be stated precisely and proved or made into a lemma.\n\nThe citation pattern is heavy on the author's own preprints ([Rui22], [Rui25], [RT24b]), but the cited results are independent and not circular. The paper is self-contained enough to check the main architecture.\n\nWho is this for? People working on motivic t-structures, relative 1-motives, or integral motives over bases. It deserves a serious referee; with the Corollary 5.4 reduction supplied, it would be a solid contribution. I would send it out rather than desk reject, and the referee should focus on that reduction.","headline":"A credible and nontrivial extension of the motivic t-structure to integral 1-motives over base schemes, held back by a reduction to prior work that the reader has to take on faith.","tokens_in":26285,"tokens_out":2635,"would_cite":true,"duration_ms":26921,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","14F20","14K15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the motivic t-structure, which truncates étale motives into past and future parts, restricts to the category of 1-motives with integral coefficients over any Q-scheme or Dedekind scheme, with heart the abelian…","keywords":["1-motives","motivic t-structure","Deligne 1-motives with torsion","étale motives","good reduction","integral coefficients","Dedekind schemes","ℓ-adic Tate modules"],"falsifier":"A concrete way to test the claim is to take a non-regular Q-scheme S and an integral 1-motive M such that the truncation τ≤0(M) computed in the Ind-1-motive category is a genuine Ind-1-motive but not an object of DM1(S,Z). Equivalently, one can look for a Deligne 1-motive over the generic point of a normal Q-scheme whose ℓ-adic Tate modules all extend to local systems but which itself fails to extend over a codimension-one point; either example would contradict Corollary 5.4 or Proposition 2.10.","tokens_in":25140,"feed_emoji":"🧩","tokens_out":11124,"duration_ms":116378,"temperature":0.7,"pith_summary":"Over a base scheme that is either of characteristic zero or a Dedekind scheme, the motivic t-structure on étale motives restricts to the subcategory of 1-motives with integral coefficients, and the pieces cut out by this structure are exactly Deligne 1-motives with torsion once residue-characteristic primes are inverted. This gives the first integral-coefficient motivic t-structure for relative 1-motives; earlier constructions over arbitrary bases worked only with rational coefficients, because the integral Artin truncation functor behaves pathologically. The paper shows the heart is abelian by comparing smooth 1-motives with Deligne 1-motives through a fully faithful restriction to the generic point and a good-reduction criterion in terms of ℓ-adic Tate modules. The result puts integral 1-motives on the same footing as the rational theory and makes the ℓ-adic realization functors exact on this subcategory.","feed_headline":"Integral motivic t-structure exists on 1-motives","feed_subtitle":"The heart is the abelian category of Deligne 1-motives with torsion over Q-schemes and Dedekind schemes.","key_machinery":"The machinery is the motivic t-structure on Ind-1-motives, generated by the images of Deligne 1-motives under the functor Φ from Deligne motives to étale Voevodsky motives, together with the structural results on Deligne 1-motives with torsion: generic-fiber full faithfulness over normal schemes and the ℓ-adic Tate-module criterion for good reduction. These reduce the comparison to the regular case, where the proof distinguishes smooth 1-motives in the heart using the motivic Picard functor ω1 and degree estimates for the localization triangle of an open immersion. Once the regular case is established, the general case follows by a p-local reduction, one prime at a time, borrowed from the earlier 0-motive construction.","core_discovery":"The central claim, Corollary 5.4, is that for S a Q-scheme or a Dedekind scheme and Λ a localization of Z, the motivic t-structure induces a t-structure on the category DM1(S,Λ) of 1-motives, compatible with pullbacks and with the ℓ-adic realization functors. For regular S and coefficients in Z′, the comparison functor ΦS gives an equivalence between the category of Deligne 1-motives with torsion and the intersection of smooth 1-motives with the heart of the t-structure, so the heart is abelian. Two structural facts about Deligne 1-motives with torsion carry the proof: over a connected normal scheme they are determined by their generic fiber, and a generic Deligne 1-motive extends to the whole scheme exactly when all its ℓ-adic Tate modules extend. The main theorem is therefore presented as a reduction of the integral problem to the rational case plus a new integral comparison in the regular setting.","pith_inferences":["Extending beyond the paper, the good-reduction criterion suggests that integral 1-motives over a normal base are determined by their generic fiber plus finitely many ℓ-adic unramifiedness conditions, a Hasse-principle-type statement not spelled out in the paper.","Extending beyond the paper, a direct p-local proof for 1-motives that avoids citing the 0-motive reduction would make Theorem 5.1 self-contained; the current proof cites that reduction.","Extending beyond the paper, whether the t-structure is compatible with the six operations, such as pushforwards and tensor products, is left open; the rational theory suggests such compatibility should hold.","Extending beyond the paper, over a base where the good-reduction hypothesis fails, such as a smooth surface over a field of positive characteristic, Proposition 2.10 predicts the heart is no longer abelian, and a counterexample there would mark the precise limit of the construction."],"forward_implications":["Every integral 1-motive over a Q-scheme or Dedekind scheme is bounded in the motivic t-structure, so truncation functors are available for 1-motives integrally, not just rationally.","The heart of the t-structure on smooth 1-motives over a regular base is the abelian category of Deligne 1-motives with torsion, giving an integral analogue of the field-case equivalence.","The ℓ-adic realization functors become t-exact on DM1(S,Λ), so truncations of a 1-motive are detected on each ℓ-adic realization.","The t-structure is compatible with pullback, so base-change maps preserve non-positive and non-negative 1-motives.","Since Deligne 1-motives with torsion form a Serre subcategory of generic-fiber Deligne 1-motives when the good-reduction hypothesis holds, the category of such motives over the base is abelian."],"supporting_citations":[{"why":"The cited reduction from arbitrary bases to regular bases in the 0-motive case is the template for the corresponding step in Corollary 5.4.","marker":"[Rui22, Theorem 4.2.7]"},{"why":"Provides the rational-coefficient good-reduction criterion and the lisse 1-motive comparison that the paper extends to Z′-coefficients.","marker":"[Haa19, Theorem 4.10]"},{"why":"Gives the rational generic-point full faithfulness that Theorem 2.8 generalizes to Deligne 1-motives with torsion over normal schemes.","marker":"[Leh19b, Proposition A.11]"},{"why":"Supplies the field-case equivalence between Deligne 1-motives with torsion and étale 1-motives, used in Proposition 4.7 and Lemma 5.2.","marker":"[BVK16, Theorem 2.1.2]"},{"why":"Gives rational t-exactness for pullbacks and boundedness of rationally geometric 1-motives, used in Propositions 4.11 and 4.12.","marker":"[Leh19a, Theorem 4.1]"},{"why":"Establishes the extension theorem for abelian schemes in characteristic zero that verifies the good-reduction hypothesis for Q-schemes.","marker":"[Gro66, Corollaire 4.2]"},{"why":"The Néron–Ogg–Shafarevich criterion that verifies the good-reduction hypothesis over Dedekind schemes.","marker":"[ST71]"},{"why":"Shows that dualizable complexes in the heart of the constructible ℓ-adic t-structure are local systems, used in Theorem 5.1.","marker":"[HRS23, Theorem 6.2]"},{"why":"The rigidity theorem for torsion étale motives, used to control torsion objects in the t-structure.","marker":"[BH21, Corollary 3.2]"}],"fun_headline_variants":["Abelian heart for Deligne 1-motives with torsion","Motivic t-structure on integral 1-motives proven","Deligne 1-motives with torsion form abelian category","Integral 1-motives get motivic t-structure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the reduction from an arbitrary Q-scheme or Dedekind scheme to the regular case, a step written down for 0-motives in an earlier paper, works unchanged for integral 1-motives; that reduction is cited rather than proved here.","fun_headline_variants_meta":{"raw":{"variants":["Abelian heart for Deligne 1-motives with torsion","Motivic t-structure on integral 1-motives proven","Deligne 1-motives with torsion form abelian category","Integral 1-motives get motivic t-structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"total_tokens":1174,"prompt_tokens":853,"completion_tokens":321,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":469,"completion_tokens_details":{"reasoning_tokens":249}},"tokens_in":469,"tokens_out":321,"duration_ms":3744,"temperature":1.0,"reasoning_tokens":249,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:21:50.764783+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete way to test the claim is to take a non-regular Q-scheme S and an integral 1-motive M such that the truncation τ≤0(M) computed in the Ind-1-motive category is a genuine Ind-1-motive but not an object of DM1(S,Z). Equivalently, one can look for a Deligne 1-motive over the generic point of a normal Q-scheme whose ℓ-adic Tate modules all extend to local systems but which itself fails to extend over a codimension-one point; either example would contradict Corollary 5.4 or Proposition 2.10.","supporting_citations":[],"review_version":1}