{"id":"cdef4e45-b505-42bb-a799-a9c50fece86d","arxiv_id":"1908.01582","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A characteristic-free derived Albanese motive is defined and shown to sit in a canonical triangle with the classical Albanese scheme, with Nisnevich descent.","lead":"This paper defines a derived Albanese motive for smooth varieties over an algebraically closed field, without inverting the field's characteristic, inside Voevodsky's category of effective Nisnevich motives. It then proves a canonical triangle relating the new object to the classical Albanese scheme, and shows the construction satisfies Nisnevich descent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Proposition 3.3 is the load-bearing step worth checking.","rationale":"I read the paper in good faith and evaluated the central proof chain. The main theorem is precisely stated, the definitions are coherent, and the proof shows no obvious circularity. The reader's verdict of ACCEPT with moderate confidence is reasonable. I agree with the reader that Proposition 3.3 is the key reduction step, but I do not have a concrete counterexample or a demonstrable gap in the descent claim, so I would not adjust the verdict. The proposed check would confirm the cited descent result in a representative case without inverting the exponential characteristic, and would either close the dependency or expose a failure in the non-proper reduction.","tokens_in":25434,"tokens_out":52901,"duration_ms":576572,"concrete_test":"Extract Proposition 3.3 and give a self-contained proof of Proposition 3.4 for a minimal nontrivial case: take X a smooth projective curve over an algebraically closed field of characteristic p, let p: X• -> X be the Cech nerve of a finite surjective wild cover such as an Artin-Schreier cover, and compute H^i_et(X, Gm) and H^i_et(X•, Gm) for i = 0, 1 via the associated spectral sequence. If the comparison maps are isomorphisms for i = 0, 1 in this case and for the prime-p torsion sheaf Gm/Gm^p, the step is consistent; if not, Theorem 5.6 and Theorem 8.15 lose their non-proper reduction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 8.15) is a functorial distinguished triangle in DM^eff(k) for every X in Sm/k. After review, I found no internal inconsistency or missing hypothesis at the level of the stated assumptions. The single most load-bearing step is Proposition 3.3: it reduces arbitrary X to a proper simplicial h-hypercover by identifying tau>=0 Hom(M(X), Z(1)[2]) with the corresponding truncation for X•. This reduction is what makes Theorem 5.6, and hence Theorem 8.15, hold for all smooth X rather than only proper ones. The proof of Proposition 3.3 depends on Proposition 3.4, whose argument requires etale cohomological descent for h-hypercovers with integral coefficients on Gm and its prime-power torsion; the paper cites [10, Prop. 5.3.3] for the torsion part. I did not find a concrete flaw in that descent argument: the rational and finite-coefficient cases are combined in the standard way, and the algebraically closed field assumption makes the tensoring argument valid. The concern is therefore one of reliance on deep descent results rather than a demonstrated gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines, for every noetherian smooth and separated scheme X over an algebraically closed field k, an integral derived Albanese object LAlb(X) in Voevodsky's category DM^eff(k) by LAlb(X) := τ_{≥0} Hom(Hom(M(X), Z(1)[2]), Z(1)[2]). The central result (Theorem 8.15) asserts a functorial distinguished triangle NS*(X)[1] -> LAlb(X) -> Alb(X) -> NS*(X)[2], and Theorem 8.17 asserts Nisnevich descent for LAlb. The proof introduces auxiliary motives M_{≥1}(X) and M_1*(X), analyzes Picard functors of proper simplicial schemes, reduces general X to proper simplicial schemes via an h-hypercover argument (Proposition 3.3), proves a structure theorem for M_1*(X)[−1] (Theorem 5.6), establishes a motivic duality for semi-abelian varieties (Propositions 7.3 and 8.3), identifies M_1*(X) with the Cartier dual of Alb^0(X), and assembles the final triangle. The paper works integrally, i.e., without inverting the exponential characteristic, which is the main advertised improvement over Barbieri-Viale and Kahn.","tokens_in":25638,"tokens_out":21906,"duration_ms":223188,"significance":"If the proof is correct, the paper gives a canonical and functorial integral 'derived Albanese' in DM^eff(k), together with a precise relation to the classical Albanese scheme and with Nisnevich descent. The construction is explicit, contains no fitted parameters, and the main theorem is falsifiable. The proof is technically ambitious and leans on deep external results (de Jong alterations, Voevodsky vanishing theorems, representability of Picard functors, h-cohomological descent); the manuscript is not fully self-contained, but its internal architecture is coherent and the main theorems are stated with enough precision to be checked. The paper would be a useful contribution to motivic homotopy theory and to the theory of 1-motives.","major_comments":[{"comment":"The step from cohomological descent for the sheaf F to the vanishing (3.4.1) is not justified as written. From H^i_et(X,F) ≅ H^i_et(X•,F) one obtains an isomorphism of the middle terms in the long exact sequences attached to 0 -> Gm --n-> Gm -> F -> 0; this constrains the cokernel of n-multiplication and the n-torsion in the cohomology of Gm, but it does not directly yield Hom_{D(k_et,Z)}(Z, K[i]) ⊗ Z/n = 0. Since Hom_{D}(Z, K[i]) is only known to be torsion after the rational step, the implication requires a diagram chase, a Bockstein/universal-coefficients argument, or an additional finiteness statement. This point is load-bearing: Proposition 3.3 is the only reduction from arbitrary X in Sm/k to proper simplicial schemes, and it feeds Theorem 5.6 and hence Theorem 8.15. Please expand this part of the proof.","section":"Proposition 3.4, Eq. (3.4.1)"}],"minor_comments":[{"comment":"In the proof of Proposition 6.4 there is an indexing mismatch: to use the t-structure one should rewrite Hom(A[i], Z(1)[2]) as Hom(A[i−1], Gm) via Z(1)[1] ≃ Gm and then use that A[i−1] is t-positive for i > 1; the displayed sentence with A[i−2] and Z(1)[1] does not directly correspond to the Hom group being considered.","section":"Proposition 6.4"},{"comment":"The identification of τ_{≥0} Hom(M_1*(X), Z(1)[2]) ⊕ M0(X) with Alb(X) silently uses the splitting of the exact sequence 0 -> Alb^0(X) -> Alb(X) -> π0(Alb(X)) -> 0. This splitting is standard because π0(Alb(X)) is a finitely generated free abelian group and k is algebraically closed, but it should be stated explicitly at the point where (8.15.4) is written down.","section":"Theorem 8.15, proof of (8.15.4)"},{"comment":"In the proof of Proposition 8.4 the expression 'f τ− τ f′' is missing a prime on the second τ and the notation f′ is introduced only implicitly; the intended equation is f τ = τ′ f′. Please correct the notation and add one sentence explaining why the displayed diagram forces the vanishing of the difference.","section":"Proposition 8.4"},{"comment":"The phrase 'Nisnevich distinguished triangle in Sm/k' is nonstandard: the diagram displayed is a homotopy cartesian square, not a triangle. Rephrasing as 'Nisnevich distinguished square' would avoid confusion.","section":"Theorem 8.17"}],"recommendation":"major_revision","confidential_remarks":"The advertised results are substantial and the overall strategy appears credible. The main revision should focus on making the descent argument in Proposition 3.4 fully rigorous; once that is clarified, the paper is a strong candidate for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper works. It does what the abstract promises—a derived Albanese in DM_eff(k) for smooth separated schemes over an algebraically closed field, defined without inverting the exponential characteristic, which is a known gap in Barbieri-Viale–Kahn—and the main theorem, the functorial triangle NS*(X)[1] → LAlb(X) → Alb(X) → NS*(X)[2], holds up on reading.\n\nWhat is genuinely new: the definition is a double internal Hom with a τ≥0 truncation, absent from [6], and the point is that it works integrally. The proof strategy is honest: the paper flags that the Nisnevich version of L → A → Pic^0_{X/k} → 0 is not a consequence of the étale version, and builds L and A by hand via de Jong alterations. Theorem 5.6 is the technical heart—killing the G_m factor by applying Hom(G_m, −) and invoking cancellation is a nice argument. No self-citations, no fitted parameters, no circularity; the classical Albanese is the target, not an input.\n\nSoft spots, in proportion. The proof is long and leans on heavy external machinery. The load-bearing step is Proposition 3.3, reducing arbitrary X to a proper simplicial h-hypercover; it depends on étale cohomological descent for h-hypercovers with integral coefficients on G_m and its prime-power torsion ([10, Prop. 5.3.3]). I went through the rational/finite-coefficient combination: it is the standard two-step argument, and the algebraically closed hypothesis makes the tensoring step valid. This is reliance on deep published results, not a demonstrated gap—but a referee should verify the cited descent theorems apply to the simplicial case as stated. Minor items: Proposition 2.11 cites a preprint ([16, Lemma 2.12]); Proposition 2.18 cites [1, Thm 5.1.1] for representability of the simplicial Picard functor, worth confirming; there are a couple of typos ('scheme group scheme', a missing verb in 5.4) and a few proofs compressed with 'argue as in'. I also checked the terse 'This induces a distinguished triangle' in the proof of 8.15—it is justified by the direct-sum decomposition of the cone. None of this affects the mathematics as far as I can see.\n\nAudience: specialists in motivic homotopy theory and 1-motives. It fills a real gap without reshaping the field. It deserves a serious referee: precise claims, checkable chain, standard tools. My recommendation: send to peer review; expect acceptance after a referee pass on Propositions 3.3–3.4 and 2.18. I agree with the reader's ACCEPT at moderate confidence, and the stress-test note's read on 3.3 matches mine.","headline":"A credible, careful paper that removes the p-inversion from the derived Albanese and proves the expected triangle; the load-bearing descent step (Prop 3.3) is heavy but checkable, not a gap.","tokens_in":26167,"tokens_out":16951,"would_cite":true,"duration_ms":152584,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","14C15","14K30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A derived Albanese motive recovers the Albanese variety integrally.","keywords":["Albanese varieties","Derived Albanese","Effective Nisnevich motives","Néron-Severi group","Picard functors","Homotopy t-structure","Cartier dual","Nisnevich descent"],"falsifier":"Exhibit a smooth $X$ over an algebraically closed field and an h-hypercover $X_\\bullet\\to X$ for which the induced map $\\tau_{\\ge 0}\\operatorname{Hom}(M(X),\\mathbb{Z}(1)[2])\\to \\tau_{\\ge 0}\\operatorname{Hom}(M(X_\\bullet),\\mathbb{Z}(1)[2])$ is not an isomorphism; equivalently, find a prime-power torsion class in the étale cohomology $H^i_{\\text{\\'et}}(X,\\mathbb{G}_m)$ that violates h-hypercover descent. Such an example would break Proposition 3.3 and with it the structure theorem for non-proper $X$ and Theorem 8.15.","tokens_in":25214,"feed_emoji":"🔺","tokens_out":10106,"duration_ms":90973,"temperature":0.7,"pith_summary":"Over an algebraically closed field, every smooth separated scheme $X$ is shown to carry a derived Albanese object $\\mathrm{LAlb}(X)$ inside Voevodsky's triangulated category of effective Nisnevich motives. The object is defined by a truncation of the double internal Hom involving the motive of $X$ and $\\mathbb{Z}(1)[2]$. The main theorem packages the Albanese variety together with the Néron–Severi group in a functorial distinguished triangle $\\mathrm{NS}^*(X)[1]\\to \\mathrm{LAlb}(X)\\to \\mathrm{Alb}(X)\\to \\mathrm{NS}^*(X)[2]$. The definition has two features that justify it: it works with integral coefficients, without inverting the exponential characteristic of the field, and it satisfies Nisnevich descent. A careful reader should care because this gives a purely motivic, characteristic-free description of a classical birational invariant.","feed_headline":"Derived Albanese motive recovers the Albanese variety integrally","feed_subtitle":"For smooth schemes, the derived Albanese differs from the Albanese variety only by a Néron–Severi motive, integrally.","key_machinery":"The load-bearing machinery is the $0$-motivic (homotopy) t-structure on $\\mathrm{DM}^{\\mathrm{eff}}(k)$, generated by motives of smooth schemes, together with its truncation functor $\\tau_{\\ge 0}$. The auxiliary object $M^*_1(X)$, defined as the cocone of the map $\\operatorname{Hom}(M_{\\ge 1}(X),\\mathbb{Z}(1)[2])\\to \\mathrm{NS}(X)$, is the workhorse of the proof. Two identifications carry the argument: $M^*_1(X)$ is isomorphic to the Cartier dual $(\\mathrm{Alb}^0(X))^\\vee$ of the Albanese variety (Theorem 8.8), and for every semi-abelian variety $G$ the truncation of $\\operatorname{Hom}(G^\\vee,\\mathbb{Z}(1)[2])$ is isomorphic to $G$ itself (Proposition 8.3). These isomorphisms let the universal property of the Albanese be read off from Hom groups in the motivic category.","core_discovery":"Theorem 8.15 is the paper's central assertion: for every $X\\in \\mathrm{Sm}/k$, there is a functorial distinguished triangle $\\mathrm{NS}^*(X)[1]\\to \\mathrm{LAlb}(X)\\to \\mathrm{Alb}(X)\\to \\mathrm{NS}^*(X)[2]$ in $\\mathrm{DM}^{\\mathrm{eff}}(k)$, where $\\mathrm{LAlb}(X):=\\tau_{\\ge 0}\\operatorname{Hom}(\\operatorname{Hom}(M(X),\\mathbb{Z}(1)[2]),\\mathbb{Z}(1)[2])$ and $\\mathrm{NS}^*(X):=\\operatorname{Hom}(\\mathrm{NS}(X),\\mathbb{Z}(1)[1])$. In prose, the derived Albanese is the truncation of the double dual of the motive of $X$, and it differs from the classical Albanese variety only by a shift of the Néron–Severi motive. The proof shows that the auxiliary motive $M^*_1(X)$, defined by the triangle $M^*_1(X)\\to \\operatorname{Hom}(M_{\\ge 1}(X),\\mathbb{Z}(1)[2])\\to \\mathrm{NS}(X)\\to M^*_1(X)[1]$, is isomorphic to the Cartier dual of the Albanese variety $\\mathrm{Alb}^0(X)$. Feeding that identification into the double-Hom construction produces the triangle. The integrality point is that an earlier derived Albanese construction only worked after inverting the exponential characteristic, whereas this definition does not require that inversion.","pith_inferences":["The same definition is likely to extend from algebraically closed fields to perfect fields once the integral étale descent statements for $\\mathbb{G}_m$ under h-covers are available; the algebraically closed hypothesis simplifies several sheaf-theoretic arguments but may not be essential to the triangle.","One could define $\\mathrm{LAlb}$ directly on simplicial schemes and ask whether the distinguished triangle and Nisnevich descent formalize into a motivic $1$-motive whose realizations recover the classical Albanese $1$-motive; the Cartier-dual identification is a first step in that direction.","The Néron–Severi term in the triangle suggests a derived-category refinement of the classical exact sequence $0\\to \\mathrm{Pic}^0(X)\\to \\mathrm{Pic}(X)\\to \\mathrm{NS}(X)\\to 0$, with the same truncation mechanism potentially yielding a motivic derived Picard object and a dual triangle."],"forward_implications":["The Albanese variety becomes a functor of the motive $M(X)$: the double-Hom construction recovers $\\mathrm{Alb}(X)$ with the Néron–Severi motive as the only correction term.","Nisnevich descent for $\\mathrm{LAlb}$ makes the derived Albanese computable by Nisnevich-local glueing, so it behaves like a motivic sheaf rather than a single birational invariant.","The isomorphisms $M^*_1(X)\\cong (\\mathrm{Alb}^0(X))^\\vee$ and $h^0(\\operatorname{Hom}(G^\\vee,\\mathbb{Z}(1)[2]))\\cong G$ give an explicit motivic model for the Cartier dual of semi-abelian varieties.","Because the construction is integral, the motivic Albanese invariant is nontrivial in positive characteristic; inverting the exponential characteristic is not needed."],"supporting_citations":[{"why":"Supplies the earlier definition of the derived Albanese; its comparison with the Albanese variety required inverting the exponential characteristic, which the present definition avoids.","marker":"[6]"},{"why":"Foundational reference for effective Nisnevich motives, including compactness of $M(X)$ and the vanishing theorem used to compute hom groups.","marker":"[17]"},{"why":"Provides representability of Picard functors and the simplicial Hilbert Theorem 90 used for the non-proper reduction.","marker":"[7]"},{"why":"Provides smooth alterations used to build a proper simplicial scheme with normal-crossing boundary over any smooth $X$.","marker":"[13]"},{"why":"Shows h-covers are universally of cohomological descent for étale sheaves of $\\mathbb{Z}/n$-modules, the integral input in the descent step.","marker":"[10]"},{"why":"Gives the isomorphism of motives of $X$ and its h-hypercover in $\\mathrm{DM}_{\\text{\\'et}}(k,\\mathbb{Q})$, used for rational-coefficient descent.","marker":"[11]"},{"why":"Introduces the Albanese scheme as the universal morphism to semi-abelian group schemes, the comparison target of the theorem.","marker":"[22]"},{"why":"Establishes duality relations between Albanese and Picard $1$-motives used to identify the connected component of the Albanese.","marker":"[21]"},{"why":"Cancellation theorem used to compute Hom groups involving $\\mathbb{G}_m$, in particular to kill the torus part in the semi-abelian reduction.","marker":"[25]"},{"why":"Provides representability and extension facts for commutative group schemes used in the semi-abelian and Picard computations.","marker":"[19]"}],"fun_headline_variants":["Derived Albanese recovers Albanese integrally, no prime inversion needed","Albanese variety from derived motive, up to a Néron–Severi shift","Derived Albanese = Albanese + Néron–Severi shift, integrally","Nisnevich descent and integral recovery for derived Albanese","Derived Albanese motive: integral recovery without characteristic inversion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing assumption is that replacing a smooth variety by a hypercover of proper schemes does not change the truncated motive built from it; this is really a statement about how the multiplicative-group sheaf and its torsion behave under such covers, and if that statement failed for some field the conclusion for non-proper varieties would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Derived Albanese recovers Albanese integrally, no prime inversion needed","Albanese variety from derived motive, up to a Néron–Severi shift","Derived Albanese = Albanese + Néron–Severi shift, integrally","Nisnevich descent and integral recovery for derived Albanese","Derived Albanese motive: integral recovery without characteristic inversion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":2997,"prompt_tokens":911,"completion_tokens":2086,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":1988}},"tokens_in":527,"tokens_out":2086,"duration_ms":17362,"temperature":1.0,"reasoning_tokens":1988,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:09:20.728099+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a smooth $X$ over an algebraically closed field and an h-hypercover $X_\\bullet\\to X$ for which the induced map $\\tau_{\\ge 0}\\operatorname{Hom}(M(X),\\mathbb{Z}(1)[2])\\to \\tau_{\\ge 0}\\operatorname{Hom}(M(X_\\bullet),\\mathbb{Z}(1)[2])$ is not an isomorphism; equivalently, find a prime-power torsion class in the étale cohomology $H^i_{\\text{\\'et}}(X,\\mathbb{G}_m)$ that violates h-hypercover descent. Such an example would break Proposition 3.3 and with it the structure theorem for non-proper $X$ and Theorem 8.15.","supporting_citations":[{"cited_title":"Barbieri-Viale and B","cited_arxiv_id":null,"evidence_quote":"Supplies the earlier definition of the derived Albanese; its comparison with the Albanese variety required inverting the exponential characteristic, which the present definition avoids."},{"cited_title":"Mazza, V","cited_arxiv_id":null,"evidence_quote":"Foundational reference for effective Nisnevich motives, including compactness of $M(X)$ and the vanishing theorem used to compute hom groups."},{"cited_title":"Barbieri-Viale and V","cited_arxiv_id":null,"evidence_quote":"Provides representability of Picard functors and the simplicial Hilbert Theorem 90 used for the non-proper reduction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides smooth alterations used to build a proper simplicial scheme with normal-crossing boundary over any smooth $X$."},{"cited_title":"Cisinski and F","cited_arxiv_id":null,"evidence_quote":"Shows h-covers are universally of cohomological descent for étale sheaves of $\\mathbb{Z}/n$-modules, the integral input in the descent step."},{"cited_title":"Cisinski and F","cited_arxiv_id":null,"evidence_quote":"Gives the isomorphism of motives of $X$ and its h-hypercover in $\\mathrm{DM}_{\\text{\\'et}}(k,\\mathbb{Q})$, used for rational-coefficient descent."},{"cited_title":"Serre , Morphismes universels et diff´ erentielles de troisi` eme es p` ece, S´ eminaire Claude Chevalley, 4 (1958-1959)","cited_arxiv_id":null,"evidence_quote":"Introduces the Albanese scheme as the universal morphism to semi-abelian group schemes, the comparison target of the theorem."},{"cited_title":"Ramachandran , Duality of Albanese and Picard 1-motives , K-theory, 22 (2001), pp","cited_arxiv_id":null,"evidence_quote":"Establishes duality relations between Albanese and Picard $1$-motives used to identify the connected component of the Albanese."},{"cited_title":"Math., (2010), pp","cited_arxiv_id":null,"evidence_quote":"Cancellation theorem used to compute Hom groups involving $\\mathbb{G}_m$, in particular to kill the torus part in the semi-abelian reduction."},{"cited_title":"Oort , Commutative group schemes , vol","cited_arxiv_id":null,"evidence_quote":"Provides representability and extension facts for commutative group schemes used in the semi-abelian and Picard computations."}],"review_version":1}