Pith. sign in

REVIEW 3 major objections 5 minor 44 references

Deligne 1-motives with torsion and \'etale motives

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.10637 v1 pith:Q5NUXLXI submitted 2025-06-12 math.AG

Raphaël Ruimy This is my paper · ORCID
classification math.AG MSC 14F4214F2014K15
keywords 1-motivesmotivict-structureDelignewithtorsionétalemotivesgoodreductionintegralcoefficientsDedekindschemesℓ-adicTatemodules
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Section 5, proof of Corollary 5.4] 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.
  2. [Section 4, Corollary 4.16] 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.
  3. [Section 4, Remark 4.14] 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.
minor comments (5)
  1. [Abstract and Introduction] There are a few typographical glitches, such as 'on1-motives' in the abstract and 'finial section' in the introduction; these should be corrected.
  2. [Section 2, proof of Proposition 2.10] 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′'.
  3. [Section 2, proof of Theorem 2.8] The reference '[Ray70, Collaire IX.1.4]' should presumably read 'Corollaire IX.1.4'.
  4. [Section 3, proof of Lemma 3.7] 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.
  5. [Section 5, proof of Theorem 5.1] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

Singular-base case of main theorem rests on a one-line reduction to the author's own [Rui22]; otherwise the derivation is substantively self-contained.

  1. self citation load bearing [Corollary 5.4, proof (Section 5)]
    "exactly as in the proof of [Rui22, Theorem 4.2.7], we can reduce to the case when 1. p is invertible on S or S is of characteristic p. 2. M belongs to DMsm1(S, Λ)."

    This is the sole step that extends the main theorem from the regular case proved in Theorem 5.1 to arbitrary Q-schemes, which is the genuinely new singular-base content of Corollary 5.4. The reduction is not proved for integral 1-motives; it is imported from the author's earlier [Rui22], which treats 0-motives over excellent schemes allowing resolution by alterations. The paper gives no argument that p-geometric Ind-1-motives, the class DMsm1, or the t-structure restriction behave as required for 1-motive generators on singular bases. The main theorem for singular Q-schemes therefore reduces to an unverified self-citation rather than to a derivation contained in this paper.

full rationale

The core of the paper is not circular by construction. The motivic t-structure on Ind-1-motives is generated by images of Deligne 1-motives (Definition 4.1), but the nontrivial content lies in proving that these generators land in the heart (Corollary 4.16), that smooth 1-motives in the heart are exactly Deligne 1-motives (Theorem 5.1), and that the t-structure restricts to DM1(S,Λ) (Corollary 5.4). Those proofs use independent ingredients: the field-case t-exactness from BVK16, rigidity, the Néron–Ogg–Shafarevich criterion and Hypothesis 2.3, and Lemma 5.3. No fitted parameter is renamed as a prediction, and no known result is repackaged as new. The single load-bearing self-citation is the reduction in the proof of Corollary 5.4, where the singular-Q-scheme case is dispatched by copying [Rui22, Theorem 4.2.7] without checking that the p-geometric reduction works for integral 1-motives. This is a self-citation carrying the central new case, but it is not a definitional tautology, so the score is moderate rather than high.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted. The new categories (MD_1(S), the motivic t-structure on Ind-1-motives) are constructions proved in the text rather than postulated entities, so no invented entities are listed. The axioms are the external or explicitly stated inputs the central proof depends on.

assumptions (5)
  • domain assumption Rigidity theorem of Bachmann-Hoyois (BH21, Cor. 3.2): for ℓ invertible on S, ρ! is an equivalence between ℓ-torsion étale sheaves and ℓ-torsion étale motives.
    Used as a black box in Lemma 4.9, Proposition 3.4, Lemma 3.7, and Corollary 4.13. Cited from an arXiv preprint, not proved in this paper.
  • domain assumption Hypothesis 2.3: an abelian variety over the generic point of a connected normal scheme with ℓ-adic Tate module unramified on S[1/ℓ] extends to an abelian scheme.
    Stated as a hypothesis; verified for Dedekind schemes by Néron-Ogg-Shafarevich and for Q-schemes by Grothendieck (Remark 2.4). Underlies Proposition 2.10 and hence Theorem 5.1.
  • domain assumption Continuity of étale motives with integral coefficients, adapting CD16, Theorem 6.3.9.
    Used in the proofs of Theorem 5.1 and Lemma 5.2 to transfer statements from the generic point to an open subscheme. The paper asserts the integral adaptation works by the same proof, without giving it.
  • ad hoc to paper Lehalleur's rational-coefficient 1-motive machinery extends to Λ⊗Q coefficients for any flat Z-algebra Λ.
    Remark 4.14 states this 'is indeed true by direct inspection of all the necessary proofs in [Leh19b]' and invites the reader to check; it is used to justify Corollary 4.13 over general Λ.
  • standard math Voevodsky's cancellation theorem for étale motives at the level used in Lemma 3.7 and Construction 4.6.
    Cited from BVK16 and Cisinski-Déglise; standard in the motivic literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Deligne 1-motives with torsion and \'etale motives." pith.science (2026). https://pith.science/paper/Q5NUXLXI

@misc{pith2026250610637,
  author       = {Pith},
  title        = {Pith review of: Deligne 1-motives with torsion and \'etale motives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5NUXLXI}},
  note         = {Machine review of arXiv:2506.10637}
}
read the original abstract

We construct the motivic t-structure on 1-motives with integral coefficients over a scheme of characteristic zero or a Dedekind scheme. When we invert the residue characteristic exponents of the base, this t-structure induces a t-structure on the category of smooth 1-motives whose heart is the category of Deligne 1-motives with torsion which we prove to be abelian. This relies on the fact that over a normal scheme, Deligne 1-motives with torsion are determined by their fiber on the generic point and on an explicit description of good reduction Deligne 1-motives with torsion.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 37 canonical work pages

  1. [1]

    1-motivic sheaves and the Albanese functor

    Joseph Ayoub and Luca Barbieri-Viale . 1-motivic sheaves and the Albanese functor. J. Pure Appl. Algebra , 213(5):809--839, 2009

    work page 2009

  2. [2]

    Th \'e orie des topos et cohomologie \'e tal des sch \'e mas.(sga4) springer lnm 269 and 270, 1972

    Michael Artin, Alexandre Grothendieck, and Jean-Louis Verdier. Th \'e orie des topos et cohomologie \'e tal des sch \'e mas.(sga4) springer lnm 269 and 270, 1972

    work page 1972

  3. [3]

    On the relative motive of a commutative group scheme

    Giuseppe Ancona, Annette Huber, and Simon Pepin Lehalleur. On the relative motive of a commutative group scheme. Algebr. Geom. , 3(2):150--178, 2016

    work page 2016

  4. [4]

    The \(n\) -motivic \(t\) -structures for \(n=0,1\) and \(2\)

    Joseph Ayoub. The \(n\) -motivic \(t\) -structures for \(n=0,1\) and \(2\) . Adv. Math. , 226(1):111--138, 2011

    work page 2011

  5. [5]

    La r \'e alisation \'e tale et les op \'e rations de Grothendieck , volume 47

    Joseph Ayoub. La r \'e alisation \'e tale et les op \'e rations de Grothendieck , volume 47. Soci \'e t \'e Math \'e matique de France, 2014

    work page 2014

  6. [6]

    Faisceaux pervers , volume 100

    Alexander Beilinson, Joseph Bernstein, Pierre Deligne, and Ofer Gabber. Faisceaux pervers , volume 100. Soci \'e t \'e math \'e matique de France, 2018

    work page 2018

  7. [7]

    Controlled objects in left-exact -categories and the novikov conjecture

    Ulrich Bunke, Denis-Charles Cisinski, Daniel Kasprowski, and Christoph Winges. Controlled objects in left-exact -categories and the novikov conjecture. Available at https://arxiv.org/abs/1911.02338, 2019

    arXiv 1911

  8. [8]

    Remarks on \'etale motivic stable homotopy theory

    Tom Bachmann and Marc Hoyois. Remarks on \'etale motivic stable homotopy theory. Availible at https://arxiv.org/abs/2104.06002, 2021

    arXiv 2021

Show all 44 references
  1. [9]

    N \'e ron models , volume 21 of Ergeb

    Siegfried Bosch, Werner L \"u tkebohmert, and Michel Raynaud. N \'e ron models , volume 21 of Ergeb. Math. Grenzgeb., 3. Folge . Berlin etc.: Springer-Verlag, 1990

    1990

  2. [10]

    On the derived category of 1-motives , volume 381 of Ast \'e risque

    Luca Barbieri-Viale and Bruno Kahn. On the derived category of 1-motives , volume 381 of Ast \'e risque . Paris: Soci \'e t \'e Math \'e matique de France (SMF), 2016

    2016

  3. [11]

    Barbieri-Viale, A

    L. Barbieri-Viale, A. Rosenschon, and M. Saito. Deligne's conjecture on 1-motives. Ann. Math. (2) , 158(2):593--633, 2003

    2003

  4. [12]

    \'E tale motives

    Denis-Charles Cisinski and Fr \'e d \'e ric D \'e glise. \'E tale motives. Compositio Mathematica , 152(3):277--427, 2016

    2016

  5. [13]

    Triangulated categories of mixed motives , volume 6 of Springer monographs in mathematics

    Denis-Charles Cisinski and Fr \'e d \'e ric D \'e glise. Triangulated categories of mixed motives , volume 6 of Springer monographs in mathematics . Springer, 2019

    2019

  6. [14]

    Higher categories and homotopical algebra , volume 180 of Camb

    Denis-Charles Cisinski. Higher categories and homotopical algebra , volume 180 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2019

    2019

  7. [15]

    Reductive group schemes

    Brian Conrad. Reductive group schemes. In Autour des sch\'emas en groupes. \'Ecole d'\'Et\'e ``Sch\'emas en groupes''. Volume I , pages 93--444. Paris: Soci \'e t \'e Math \'e matique de France (SMF), 2014

    2014

  8. [16]

    Lectures on condensed mathematics, 2021

    Dustin Clausen and Peter Scholze. Lectures on condensed mathematics, 2021. Available at https://www.math.uni-bonn.de/people/scholze/Condensed.pdf

    2021

  9. [17]

    Th \'e orie de Hodge

    Pierre Deligne. Th \'e orie de Hodge . III . Publ. Math., Inst. Hautes \'E tud. Sci. , 44:5--77, 1974

    1974

  10. [18]

    Artin, J.-E

    Michel Demazure, Alexander Grothendieck, M. Artin, J.-E. Bertin, Pierre Gabriel, Michel Raynaud, and Jean-Pierre Serre, editors. S \'e minaire de g \'e om \'e trie alg \'e brique du Bois Marie 1962-64. Sch \'e mas en groupes ( SGA 3). Springer, 1970

    1962

  11. [19]

    Rev \^e tement \'e tales et groupe fondamental S \'e minaire de G \'e ometrie Alg \'e brique du Bois-Marie 1960-1961 (SGA 1) , volume 224

    Alexander Grothendieck and Michèle Raynaud. Rev \^e tement \'e tales et groupe fondamental S \'e minaire de G \'e ometrie Alg \'e brique du Bois-Marie 1960-1961 (SGA 1) , volume 224. Springer, 1971

    1960

  12. [20]

    \'E l \'e ments de g \'e om \'e trie alg \'e brique

    Alexander Grothendieck. \'E l \'e ments de g \'e om \'e trie alg \'e brique. I : Le langage des sch \'e mas. Publ. Math., Inst. Hautes \'E tud. Sci. , 4:1--228, 1960

    1960

  13. [21]

    Un th \'e or \`e me sur les homomorphismes de schemas abeliens

    Alexander Grothendieck. Un th \'e or \`e me sur les homomorphismes de schemas abeliens. Invent. Math. , 2:59--78, 1966

    1966

  14. [22]

    \'E l \'e ments de g \'e om \'e trie alg \'e brique

    Alexander Grothendieck. \'E l \'e ments de g \'e om \'e trie alg \'e brique. IV : \'E tude locale des sch \'e mas et des morphismes de sch \'e mas. R \'e dig \'e avec la colloboration de Jean Dieudonn \'e . Publ. Math., Inst. Hautes \'E tud. Sci. , 32:1--361, 1967

    1967

  15. [23]

    Lisse 1-Motives

    Johann Haas. Lisse 1-Motives . PhD thesis, Universit \"a t Regensburg, 2019

    2019

  16. [24]

    Constructible sheaves on schemes, 2023

    Tamir Hemo, Timo Richarz, and Jakob Scholbach. Constructible sheaves on schemes, 2023. Available at https://arxiv.org/pdf/2305.18131

    2023 arXiv

  17. [25]

    On the arithmetic of 1-motives

    Peter Jossen. On the arithmetic of 1-motives . PhD thesis, Central European University, 2009

    2009

  18. [26]

    Space filling curves over finite fields

    Nicholas M Katz. Space filling curves over finite fields. Mathematical Research Letters , 6(6):613--624, 1999

    1999

  19. [27]

    Constructible \(1\) -motives and exactness of realisation functors

    Simon Pepin Lehalleur. Constructible \(1\) -motives and exactness of realisation functors. Doc. Math. , 24:1721--1737, 2019

    2019

  20. [28]

    Triangulated categories of relative 1-motives

    Simon Pepin Lehalleur. Triangulated categories of relative 1-motives. Advances in Mathematics , 347:473--596, 2019

    2019

  21. [29]

    Higher topos theory

    Jacob Lurie. Higher topos theory . Princeton University Press, 2009

    2009

  22. [30]

    Higher algebra, 2017

    Jacob Lurie. Higher algebra, 2017. Available at https://www.math.ias.edu/ lurie/papers/HA.pdf

    2017

  23. [31]

    Isomotifs de dimension inf \'e rieure ou \'e gale \`a un

    Fabrice Orgogozo. Isomotifs de dimension inf \'e rieure ou \'e gale \`a un . manuscripta mathematica , 115(3):339--360, 2004

    2004

  24. [32]

    Ho Quoc and Penghui Li

    P. Ho Quoc and Penghui Li. Revisiting mixed geometry. Preprint, available at http://arxiv.org/abs/2202.04833, 2023

    2023

  25. [33]

    Faisceaux amples sur les sch \'e mas en groupes et les espaces homog \`e nes , volume 119 of Lect

    Michel Raynaud. Faisceaux amples sur les sch \'e mas en groupes et les espaces homog \`e nes , volume 119 of Lect. Notes Math. Springer, Cham, 1970

    1970

  26. [34]

    Chow k \"u nneth decomposition for \'e tale motives, 2022

    Ivan Rosas Soto. Chow k \"u nneth decomposition for \'e tale motives, 2022. Available at https://arxiv.org/abs/2403.00159

    2022 arXiv

  27. [35]

    \'E tale motives of geometric origin

    Rapha \"e l Ruimy and Swann Tubach. \'E tale motives of geometric origin. Preprint, available at https://arxiv.org/abs/2405.07095, 2024

    2024 arXiv

  28. [36]

    Nori motives (and mixed Hodge modules) with integral coefficients

    Rapha \"e l Ruimy and Swann Tubach. Nori motives (and mixed Hodge modules) with integral coefficients. Preprint, available at https://arxiv.org/abs/2407.01462, 2024

    2024 arXiv

  29. [37]

    Integral A rtin motives I : S mooth objects and the ordinary t-structure, 2022

    Raphaël Ruimy. Integral A rtin motives I : S mooth objects and the ordinary t-structure, 2022. Available at https://arxiv.org/abs/2211.02505

    2022 arXiv

  30. [38]

    Artin perverse sheaves

    Raphaël Ruimy. Artin perverse sheaves. Journal of Algebra , 639:596--677, 2024

    2024

  31. [39]

    Integral A rtin motives II : P erverse motives and A rtin V anishing T heorem, 2025

    Raphaël Ruimy. Integral A rtin motives II : P erverse motives and A rtin V anishing T heorem, 2025. Available at https://arxiv.org/abs/2504.15732

    2025 arXiv

  32. [40]

    Good reduction of abelian varieties

    Jean-Pierre Serre and John Tate. Good reduction of abelian varieties. Matematika, Moskva 15, No . 5, 140-165 (1971)., 1971

    1971

  33. [41]

    The stacks project

    The Stacks project authors . The stacks project. Available at https://stacks.math.columbia.edu, 2025

    2025

  34. [42]

    Artin motives in relative Nori and Voevodsky motives

    Swann Tubach. Artin motives in relative Nori and Voevodsky motives. Preprint, available at http://swann.tubach.fr, 2024

    2024

  35. [43]

    Punctual gluing of t-structures and weight structures

    Vaibhav Vaish. Punctual gluing of t-structures and weight structures. manuscripta mathematica , pages 1--26, 2019

    2019

  36. [44]

    Homology of schemes and covariant motives

    Vladimir Voevodsky. Homology of schemes and covariant motives . ProQuest LLC, Ann Arbor, MI, 1992. Thesis (Ph.D.) Harvard University

    1992

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.