Pith. sign in

REVIEW 3 major objections 6 minor 64 references

A motivic approach to rational $p$-adic cohomologies

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read P-adic weight-monodromy proved for toric complete intersections

desk verdict A well-flagged survey plus one new proof route whose linchpin—the compatibility of the motivic realization with Hyodo-Kato—needs expansion before it can be independently checked. read the letter →

arxiv 2508.16196 v1 pith:OSU5I532 submitted 2025-08-22 math.AG

Federico Binda , Alberto Vezzani This is my paper · ORCID
classification math.AG MSC 14F3014F4214G2214M25
keywords p-adiccohomologyweight-monodromyconjecturemotivichomotopytheoryrigidanalyticmotivesHyodo-Katonearbycyclestiltingequivalencetoricvarieties
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

This paper argues that motivic homotopy theory gives a canonical home for p-adic cohomology: rigid/overconvergent cohomology in characteristic p and Hyodo-Kato cohomology in mixed characteristic both arise as realizations of motives, with monodromy encoded by an equivalence between analytic motives over K and algebraic motives over the residue field equipped with a nilpotent operator. The new result is a proof of the p-adic weight-monodromy conjecture for smooth scheme-theoretic complete intersections inside projective smooth toric varieties over a finite extension of Qp: on Hyodo-Kato cohomology, Frobenius acts purely on each graded piece of the monodromy filtration. The proof transfers purity from the tilt (the characteristic-p world) to the original variety by showing the cohomological motive of the complete intersection is a direct summand of the motive of a smooth alteration, and it replaces an earlier log-geometry compatibility check with a detection principle built from the motivic equivalence. If correct, the same motivic mechanism yields p-adic Clemens-Schmid sequences and a uniform treatment of nearby cycles across l-adic, p-adic, and complex Hodge realizations.

What carries the argument

The load-bearing object is the monodromy category DA(k)^N: algebraic motives over the residue field k equipped with a locally nilpotent endomorphism N: M -> M(-1) (a monodromy operator twisting by -1). Theorem 4.5 (from [14]) identifies this category with RigDA(K)^gr, the essential image of the generic-fiber functor from algebraic motives over k to rigid analytic motives over K, depending on a pseudo-uniformizer. This equivalence simultaneously defines the motivic nearby cycle Psi M and its monodromy N, and it allows any characteristic-p realization F: DA(k) -> D to be extended to a monodromy-carrying realization on RigDA^gr(K) (the detection principle, Prop. 4.17); the extension is classifi

What would settle it

Compute, for the rigid analytic quotient G_m^an/ϖ^Z (the elliptic curve with nodal reduction), the composite motivic realization into (φ,N)-modules: it must equal the classical object, whose monodromy operator is a single off-diagonal 1 between the unit twist and its twist by −1. A mismatch in the monodromy or Frobenius action would falsify the functor identification on which Corollary 4.32 rests. Alternatively, for a smooth complete intersection with semistable reduction, compute the Frobenius eigenvalues on gr^M_j H^i_HK and check they all have Weil weight i+j.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the motivic equivalence between rigid analytic motives over a non-archimedean field K and algebraic motives over its residue field k equipped with a locally nilpotent monodromy operator (Theorem 4.5) lets one read the classical Hyodo-Kato realization as the extension of a characteristic-p motivic realization, and thereby transfer purity statements from characteristic p to mixed characteristic. Concretely, Corollary 4.32 establishes that for Y a smooth scheme-theoretic complete intersection in a projective smooth toric variety over a finite extension of Qp, the Frobenius action on the j-th graded quotient of the monodromy filtration on Hyodo-Kato cohomolo

Load-bearing premise

The whole transfer rests on the one-sentence assertion that the motivic realization functor built from the equivalence RigDA^gr ≃ DA^N is exactly the classical Hyodo-Kato functor defined via logarithmic geometry (via Corollary 4.19); if that identification failed, purity would be proved for a motivic variant, not for the classical (φ,N)-modules, and Corollary 4.32 would not follow.

Editorial extensions

If this is right

  • If Corollary 4.32 is correct, the p-adic weight-monodromy conjecture holds for every smooth scheme-theoretic complete intersection in a projective smooth toric variety over a finite extension of Qp: Frobenius on gr^M_j H^i_HK is pure of weight i+j.
  • Hyodo-Kato cohomology gains a purely motivic definition, RGamma_rig(Psi X), so it carries a canonical weight filtration and fits into a Clemens-Schmid chain complex for any proper semistable scheme over the valuation ring.
  • The same construction yields a p-adic analogue of the local invariant cycle theorem: the specialization map from rigid cohomology of the special fiber to Hyodo-Kato cohomology of the generic fiber is part of an exact sequence whenever weight-monodromy holds.
  • Relative rigid cohomology obtains h-descent and its direct images are overconvergent F-isocrystals, confirming a previously open conjecture beyond the liftable case.
  • The motivic formulation makes the nearby-cycle functor independent of log-geometry choices, so the same proof template applies to other realizations, not only Hyodo-Kato cohomology.

Reading between the lines

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

  • The direct-summand mechanism should extend beyond toric complete intersections: any smooth projective variety whose motive is a self-dual ring object and whose tilted model has a smooth alteration with a non-zero top-degree map would inherit purity the same way, provided the relevant approximation theorem holds.
  • Because the motivic nearby-cycle functor is defined before realization, the weight and monodromy filtrations on Hyodo-Kato cohomology are plausibly canonical and functorial in the motive; one testable consequence is that the p-adic Clemens-Schmid maps are independent of the choice of tubular neighborhood.
  • The detection principle identifies a realization by testing one object (the elliptic curve with nodal reduction), so the same one-object check should identify other p-adic cohomology functors, such as prismatic or A_inf variants, with the classical Hyodo-Kato functor.
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 / 6 minor

Summary. The paper is a survey of motivic homotopy-theoretic methods for rational p-adic cohomology, collecting results from previous work of the authors and others, and presenting a new proof of the p-adic weight-monodromy conjecture for smooth complete intersections in projective toric varieties. The main new contribution is the proof of Corollary 4.32, which uses the equivalence RigDA(K)^gr ≃ DA(k)^N (Theorem 4.5, from [14]) and a detection principle instead of the log-geometry tilting compatibility used in [15]. The paper also reviews the construction of motivic nearby cycles, the resulting Clemens-Schmid complexes, and applications to rigid cohomology. The weight-monodromy theorem itself is already established in [15]; the paper's novel content is the alternative proof of that theorem.

Significance. If the proposed proof is completed, it is a worthwhile contribution: it recasts a key comparison from log-geometry as a structural statement about motivic categories and clarifies the role of the detection principle. The survey portions are useful and clearly organized. The theorem being proved is already known from [15], so the scientific claim of Corollary 4.32 is not at risk; what is at stake is the validity of the claimed new proof. The decisive step of that proof, however, is currently supported by an unproved compatibility assertion (Corollaries 4.19 and 4.20), and the proof of Proposition 4.31 is also abbreviated. These gaps are repairable in revision but they are load-bearing for the paper's stated original contribution.

major comments (3)
  1. [Corollary 4.32, Step (1)] Step (1) is the linchpin of the entire transfer of purity. The composite RigDAgr(L) → RigDAgr(K♭) ≃ RigDAgr(K) → D_{φ,N}(W(k)[1/p]) is asserted to 'must agree' with the Lazda–Pal Hyodo–Kato realization functor 'by means of Corollary 4.19'. But Corollary 4.19 is not proved in the text: its proof consists of the sentence 'Testing the image of this motive via classical realization functors, one can then prove...' and no compatibility is exhibited for a general object of RigDAgr(K). If this compatibility fails, the direct-summand argument applies to the wrong (φ,N)-module and the weight-monodromy conclusion for Z does not follow. This needs a full proof or an explicit published reference.
  2. [Corollary 4.20] The proof of Corollary 4.20 says the tilting equivalence is compatible with the realizations because 'it can be obtained via the equivalences RigDAgr(K) ≃ DAN(k) ≃ RigDAgr(K♭)'. Composing the two equivalences gives an equivalence of categories, but not automatically a comparison of the realizations with the classical Hyodo–Kato functors on both sides; this is exactly the compatibility statement needed in Step (1) of Corollary 4.32. The proof must identify the natural transformations between the two realization functors under the tilting equivalence, or refer to a previously proved comparison. As written, the assertion is not established.
  3. [Proposition 4.31] Proposition 4.31 is used to turn the non-zero class in CH^d into a direct-summand statement. Its proof is too compressed: the displayed weight complex is not fully specified (differentials, indexing, and the terms with |I|=2 are not analyzed), and the conclusion that Hom(1(d)[2d], M(X)) is the kernel of the 'difference' map requires checking the differentials of the complex. As it stands this is a gap in the direct-summand step, even if less central than Step (1).
minor comments (6)
  1. [Abstract / Introduction] The abstract and the opening of §1.3 refer to 'smooth projective hypersurfaces', while Theorem 1.7 and Corollary 4.32 state the result for smooth scheme-theoretic complete intersections in toric varieties. The statements should be aligned.
  2. [Corollary 4.32 proof] In the final paragraph of the proof, the notation switches from Z to X ('Mcoh(X)', 'H^*_HK(X)') without defining X. The proof should be consistent: either keep Z or explicitly set X = Z.
  3. [§2.4] The phrase 'universal homomorphism' appears where 'universal homeomorphism' is meant. Please correct.
  4. [Definition 4.1] There is a typo: 'obvsious' should be 'obvious'.
  5. [Corollary 4.25 proof] The sentence 'XΣ (and the map φ as well) has a smooth proper model XΣ over OK' overloads the notation XΣ. Use two different symbols for the scheme and its formal model.
  6. [General] Several other typos occur, e.g. 'resuls', 'Moroever', 'Clemens-Schimd', 'intepretation'. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the decisive compatibility is imported as an independent theorem from [14], anchored to Lazda–Pal's classical Hyodo–Kato realization, not derived from the target conclusion; the one-sentence assertion in Step (1) is a cited theorem, not an identity by construction.

full rationale

The proof of Corollary 4.32 reduces the weight-monodromy conjecture for Z to three ingredients: (i) a non-zero duality map obtained after tilting and alteration; (ii) Proposition 4.31, which upgrades this to a direct-summand statement; and (iii) the base case for T^flat, taken from Crew and Lazda–Pal. The only step that could look circular is Step (1) of the proof: "The composite realization RigDA^gr(L) → RigDA^gr(K^flat) ≃ RigDA^gr(K) → D_{φ,N}(W(k)[1/p]) must agree with the Hyodo–Kato realization functor defined by Lazda–Pal [43, Theorem 2.52] by means of Corollary 4.19." This is not an identity by construction; it is an application of Corollary 4.19/4.20 from the authors' earlier work [14]. Corollary 4.19 is justified there by "testing the image of this motive via classical realization functors" (the Kummer motive), and is anchored to classical Hyodo–Kato theory via [43] and to the log-geometric compatibility of [15]. Corollary 4.20's one-line proof reduces compatibility to Corollary 4.19, not to the weight-monodromy conjecture. Neither of these prior results assumes the target theorem for complete intersections; they are parameter-free comparison theorems about motivic realizations. The remaining argument is independent of these comparisons: Proposition 4.31 is proved in the text using a weight complex cited from [14, Example 4.72], and the non-vanishing of the relevant map is checked through Chow groups. There is undoubtedly a heavy self-citation chain ([14] and [15] share authors with this paper), but under the stated rules a published, parameter-free result with external anchors counts as independent support and does not constitute circularity. The real weakness is self-containment and rigor: Step (1) is asserted in one sentence, and the paper does not display a general-object comparison between the motivic composite and the classical Hyodo–Kato functor. If Corollary 4.19 or 4.20 failed, the direct-summand argument would attach purity to the wrong (φ,N)-modules. That is a correctness or completeness risk, not a circular reduction.

Assumptions & free parameters 0 free parameters · 11 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities: every category, functor and equivalence it uses is imported from prior literature, mostly the authors' own [10, 14, 15, 29, 44, 59]. The listing above collects the unproved background results the paper relies on. The load-bearing inputs for the original proof of Cor 4.32 are Theorem 4.5 ([14, Cor 4.14]) and the realization-compatibility Cor 4.19-4.20 ([14]), together with Scholze's Theorem 4.28 and the Crew/Lazda-Pal weight-monodromy theorem as external anchors.

assumptions (11)
  • domain assumption Six-functor formalism for DA(S) and RigDA(S), Q-linear, including f*, f!, localization, purity, base change (Ayoub [4,5]; Ayoub, Gallauer, Vezzani [10]; Cisinski-Deglise [20])
    Invoked throughout Section 2.3 to define homological and cohomological motives, localization triangles, and duality; used in proofs of Cor 2.21, 2.22, 4.25, and in Construction 4.30 for the self-duality of M = M^coh(X).
  • domain assumption Universal homeomorphisms induce equivalences of motivic categories: DA(X) ≃ DA(X') and RigDA(X) ≃ RigDA(X') (Theorem 2.13, from [6, Thm 1.2] and [10, Cor 2.9.11])
    Yields the Frobenius enrichment (Cor 2.16) and the equivalence M^coh(XΣ∞) ≃ M^coh(XΣ) in Cor 4.25, a key step toward the diagram in Construction 4.29.
  • domain assumption Formal and special-fiber motives agree: FDA(S) ≃ DA(Sσ) (Theorem 3.1, [10, Thm 3.1.10]); log version logFDA(S) ≃ logDA(Sσ) (Theorem 3.3, [15, Thm 2.22])
    Bridges characteristic p and mixed characteristic; underlies the functor ξ and the log Monsky-Washnitzer functor of Remark 4.14 used in the compatibility diagram with tilting.
  • domain assumption Dagger and rigid analytic motives agree: RigDA(S†) ≃ RigDA(S) (Theorem 3.5, [59, Thm 4.23])
    Allows the overconvergent de Rham realization (Prop 3.9) and the definition of rigid cohomology functors RΓrig in Section 3.3.
  • domain assumption Theorem 4.5 (= [14, Cor 4.14]): for K complete non-archimedean with perfect residue field k and value group of rational rank 1, there is an equivalence RigDA(K)gr ≃ DA(k)^N, depending on a pseudo-uniformizer
    The backbone of Section 4: yields the motivic nearby cycle (ΨM, NM), the tilting equivalence Cor 4.11, the detection principle (4.16, Prop 4.17), Cor 4.19-4.20, and the proof of Cor 4.32.
  • domain assumption Tilting equivalence for perfectoid spaces: RigDA(P) ≃ RigDA(P♭), compatible with pull-backs (Theorem 4.12, [44, Thm 5.13])
    Used in Construction 4.29 to identify M^coh(XΣ∞) with M^coh(X♭Σ∞)^♯ and to transfer Z to Z♭.
  • domain assumption Scholze's perfectoid approximation: for a hypersurface Z in a smooth proper toric variety XΣ over a perfectoid field K, there is a tilted hypersurface Z♭ and tubular neighborhoods Y, Y♭ matching under the tilting homeomorphism (Theorem 4.28, [54, Prop 8.7, Cor 8.8])
    External geometric input that produces the diagram in Construction 4.29, hence the map M^coh(Z^an) → M^coh(Z^♭an)^♯.
  • domain assumption Weight-monodromy for smooth proper varieties over Laurent series fields, in the Hyodo-Kato setting (Crew [24], Lazda-Pal [43, Thm 5.58], building on Deligne [27])
    The purity conclusion for H*_HK(Z) is inherited from H*_HK(T♭), which satisfies WM by these results; used in the final step of Cor 4.32.
  • standard math Voevodsky: motivic cohomology is isomorphic to higher Chow groups (Voevodsky [64, Cor 2])
    Used in the proof of Prop 4.31 to compute Hom(1(d)[2d], M(DI)) ≃ CH^{1-d+r}(DI)_Q and to conclude vanishing when 1-d+r < 0.
  • standard math Fulton, Intersection Theory, Lemma 12.1: deg(h!(c1(L)^d)) ≠ 0 for L ample on X♭Σ
    Used in step (3) of the proof of Cor 4.32 to show the composition CH^d(X♭Σ) → CH^d(T♭) → Z is non-zero, hence the diagonal motive map is non-zero.
  • standard math Existence of smooth alterations (de Jong [25])
    Used in Construction 4.29 and Cor 4.32 to replace the tilted complete intersection Z♭ by a smooth proper alteration T♭.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A motivic approach to rational $p$-adic cohomologies." pith.science (2026). https://pith.science/paper/OSU5I532

@misc{pith2026250816196,
  author       = {Pith},
  title        = {Pith review of: A motivic approach to rational $p$-adic cohomologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OSU5I532}},
  note         = {Machine review of arXiv:2508.16196}
}
abstract

We survey over some recent applications of motivic homotopy theory in the definition and the study of $p$-adic cohomology theories. In particular, we revisit the proof of the $p$-adic weight-monodromy conjecture for smooth projective hypersurfaces in light of the motivic definition of nearby cycles and monodromy operators.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 60 canonical work pages

  1. [15]

    BINDA , H

    F. BINDA , H. K ATO, AND A. V EZZANI , On the p-adic weight-monodromy conjecture for complete intersec- tions in toric varieties, 2022. https://arxiv.org/abs/2207.00369

    arXiv 2022

  2. [14]

    BINDA , M

    F. BINDA , M. G ALLAUER , AND A. V EZZANI , Motivic monodromy and p-adic cohomology theories. J. Eur. Math. Soc. (JEMS), Online First, 2025

    work page 2025

  3. [44]

    L E BRAS AND A

    A.-C. L E BRAS AND A. V EZZANI , The de Rham–Fargues–Fontaine cohomology, Algebra Number Theory, 17 (2023), pp. 2097–2150

    work page 2023

  4. [59]

    V EZZANI , The Monsky-Washnitzer and the overconvergent realizations , Int

    A. V EZZANI , The Monsky-Washnitzer and the overconvergent realizations , Int. Math. Res. Not. IMRN, (2018), pp. 3443–3489

    work page 2018

  5. [29]

    ERTL AND A

    V. ERTL AND A. V EZZANI , Berthelot’s conjecture via homotopy theory. Duke Math. Journal, to appear., 2024

    work page 2024

  6. [1]

    Tome 2 , Lecture Notes in Mathematics, V ol

    Th´eorie des topos et cohomologie ´etale des sch ´emas. Tome 2 , Lecture Notes in Mathematics, V ol. 270, Springer-Verlag, Berlin, 1972. S ´eminaire de G ´eom´etrie Alg ´ebrique du Bois-Marie 1963–1964 (SGA 4), Dirig´e par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. 25

    work page 1972

  7. [2]

    A NDREYCHEV , Pseudocoherent and perfect complexes and vector bundles on analytic adic spaces

    G. A NDREYCHEV , Pseudocoherent and perfect complexes and vector bundles on analytic adic spaces . arXiv:2105.12591 [math.AG], 2021

    arXiv 2021

  8. [3]

    A RABIA , Rel`evements des alg `ebres lisses et de leurs morphismes , Comment

    A. A RABIA , Rel`evements des alg `ebres lisses et de leurs morphismes , Comment. Math. Helv., 76 (2001), pp. 607–639

    work page 2001

Show all 64 references
  1. [4]

    A YOUB , Les six op ´erations de Grothendieck et le formalisme des cycles ´evanescents dans le monde motivique, I, Ast´erisque, (2007), pp

    J. A YOUB , Les six op ´erations de Grothendieck et le formalisme des cycles ´evanescents dans le monde motivique, I, Ast´erisque, (2007), pp. x+466 pp. (2008)

    2007

  2. [5]

    vi+364 pp

    , Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents dans le monde motivique, II, Ast´erisque, (2007), pp. vi+364 pp. (2008)

    2007

  3. [6]

    , La r´ealisation ´etale et les op ´erations de Grothendieck, Ann. Sci. ´Ec. Norm. Sup ´er. (4), 47 (2014), pp. 1–145

    2014

  4. [7]

    Reine Angew

    , L’alg`ebre de Hopf et le groupe de Galois motiviques d’un corps de caract´eristique nulle, I, J. Reine Angew. Math., 693 (2014), pp. 1–149

    2014

  5. [8]

    , Motifs des vari´et´es analytiques rigides, M´em. Soc. Math. Fr. (N.S.), (2015), pp. vi+386

    2015

  6. [9]

    , Weil cohomology theories and their motivic Hopf algebroid . Indag. Math. to appear. https:// arxiv.org/abs/2312.11906, 2024

    2024 arXiv

  7. [10]

    AYOUB , M

    J. AYOUB , M. G ALLAUER , AND A. V EZZANI , The six-functor formalism for rigid analytic motives, Forum Math. Sigma, 10 (2022), p. Paper No. e61

    2022

  8. [11]

    B AMBOZZI AND A

    F. B AMBOZZI AND A. V EZZANI , Rigidity for rigid analytic motives , J. Inst. Math. Jussieu, 20 (2021), pp. 1341–1369

    2021

  9. [12]

    V. G. B ERKOVICH , Smooth p-adic analytic spaces are locally contractible , Invent. Math., 137 (1999), pp. 1–84

    1999

  10. [13]

    BERTHELOT , G´eom´etrie rigide et cohomologie des vari ´et´es alg´ebriques de caract´eristique p, in Intro- ductions aux cohomologies p-adiques, D

    P. BERTHELOT , G´eom´etrie rigide et cohomologie des vari ´et´es alg´ebriques de caract´eristique p, in Intro- ductions aux cohomologies p-adiques, D. Barsky and P. Robba, eds., no. 23 in M ´emoires de la Soci ´et´e Math´ematique de France, Soci´et´e math´ematique de France, 1...

    1986

  11. [16]

    B INDA , D

    F. B INDA , D. P ARK , AND P. A. Ø STVÆR , Triangulated categories of logarithmic motives over a field , Ast´erisque, (2022), pp. ix+267

    2022

  12. [17]

    C ARO, Sur la pr´eservation de la surconvergence par l’image directe d’un morphisme propre et lisse, Ann

    D. C ARO, Sur la pr´eservation de la surconvergence par l’image directe d’un morphisme propre et lisse, Ann. Sci. ´Ec. Norm. Sup´er. (4), 48 (2015), pp. 131–169

    2015

  13. [18]

    C HIARELLOTTO , Rigid cohomology and invariant cycles for a semistable log scheme, Duke Math

    B. C HIARELLOTTO , Rigid cohomology and invariant cycles for a semistable log scheme, Duke Math. J., 97 (1999), pp. 155–169

    1999

  14. [19]

    C ISINSKI AND F

    D.-C. C ISINSKI AND F. D ´EGLISE , Mixed Weil cohomologies, Adv. Math., 230 (2012), pp. 55–130

    2012

  15. [20]

    C ISINSKI AND F

    D.-C. C ISINSKI AND F. D ´EGLISE , Triangulated categories of mixed motives , Springer Monographs in Mathematics, Springer, Cham, [2019] ©2019

    2019

  16. [21]

    C. H. C LEMENS , Degeneration of K¨ahler manifolds, Duke Math. J., 44 (1977), pp. 215–290

    1977

  17. [22]

    COLMEZ AND W

    P. COLMEZ AND W. A. NIZIOŁ , On p-adic comparison theorems for rigid analytic varieties, I, M¨unster J. Math., 13 (2020), pp. 445–507

    2020

  18. [23]

    D. A. C OX, J. B. L ITTLE , AND H. K. S CHENCK , Toric varieties, vol. 124 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2011

    2011

  19. [24]

    C REW, Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve , Ann

    R. C REW, Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve , Ann. Sci. ´Ecole Norm. Sup. (4), 31 (1998), pp. 717–763

    1998

  20. [25]

    A. J. DE JONG, Smoothness, semi-stability and alterations, Inst. Hautes ´Etudes Sci. Publ. Math., (1996), pp. 51–93

    1996

  21. [26]

    DELIGNE , Th´eorie de Hodge

    P. DELIGNE , Th´eorie de Hodge. I, in Actes du Congr`es International des Math´ematiciens (Nice, 1970), Tome 1, Gauthier-Villars ´Editeur, Paris, 1971, pp. 425–430

    1970

  22. [27]

    II, Inst

    , La conjecture de Weil. II, Inst. Hautes ´Etudes Sci. Publ. Math., (1980), pp. 137–252

    1980

  23. [28]

    E LKIK , Solutions d’´equations `a coefficients dans un anneau hens´elien, Ann

    R. E LKIK , Solutions d’´equations `a coefficients dans un anneau hens´elien, Ann. Sci. ´Ecole Norm. Sup. (4), 6 (1973), pp. 553–603

    1973

  24. [30]

    ERTL AND K

    V. ERTL AND K. YAMADA , Rigid analytic reconstruction of Hyodo–Kato theory. Preprint arXiv:1907.10964, 2019

    1907 arXiv

  25. [31]

    F LACH AND B

    M. F LACH AND B. M ORIN , Weil-´etale cohomology and zeta-values of proper regular arithmetic schemes, Doc. Math., 23 (2018), pp. 1425–1560. 26

    2018

  26. [32]

    F UJIWARA AND F

    K. F UJIWARA AND F. KATO, Foundations of rigid geometry. I, EMS Monographs in Mathematics, European Mathematical Society (EMS), Z¨urich, 2018

    2018

  27. [33]

    FULTON , Intersection theory, vol

    W. FULTON , Intersection theory, vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Springer-Verlag, Berlin, second ed., 1998

    1998

  28. [34]

    G ROSSE -K L ¨ONNE , Frobenius and monodromy operators in rigid analysis, and Drinfeld’s symmetric space, J

    E. G ROSSE -K L ¨ONNE , Frobenius and monodromy operators in rigid analysis, and Drinfeld’s symmetric space, J. Algebraic Geom., 14 (2005), pp. 391–437

    2005

  29. [35]

    G UILL ´EN AND V

    F. G UILL ´EN AND V. NAVARRO AZNAR , Sur le th ´eor`eme local des cycles invariants , Duke Math. J., 61 (1990), pp. 133–155

    1990

  30. [36]

    H UBER , ´Etale cohomology of rigid analytic varieties and adic spaces , Aspects of Mathematics, E30, Friedr

    R. H UBER , ´Etale cohomology of rigid analytic varieties and adic spaces , Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996

    1996

  31. [37]

    H YODO AND K

    O. H YODO AND K. K ATO, Semi-stable reduction and crystalline cohomology with logarithmic poles , Ast´erisque, 223 (1994), pp. 221–268

    1994

  32. [38]

    I LLUSIE , Autour du th ´eor`eme de monodromie locale , no

    L. I LLUSIE , Autour du th ´eor`eme de monodromie locale , no. 223, 1994, pp. 9–57. P ´eriodes p-adiques (Bures-sur-Yvette, 1988)

    1994

  33. [39]

    JANNSEN , On the l-adic cohomology of varieties over number fields and its Galois cohomology, in Galois groups over Q (Berkeley, CA, 1987), vol

    U. JANNSEN , On the l-adic cohomology of varieties over number fields and its Galois cohomology, in Galois groups over Q (Berkeley, CA, 1987), vol. 16 of Math. Sci. Res. Inst. Publ., Springer, New York, 1989, pp. 315–360

    1987

  34. [40]

    K ATO, Logarithmic structures of Fontaine-Illusie, in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ

    K. K ATO, Logarithmic structures of Fontaine-Illusie, in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224

    1988

  35. [41]

    K ELLY AND M

    S. K ELLY AND M. M ORROW , K-theory of valuation rings, Compos. Math., 157 (2021), pp. 1121–1142

    2021

  36. [42]

    K OSHIKAWA , Logarithmic prismatic cohomology, I

    T. K OSHIKAWA , Logarithmic prismatic cohomology, I. Preprint arXiv:2007.14037, 2022

    2007 arXiv

  37. [43]

    L AZDA AND A

    C. L AZDA AND A. P ´AL, Rigid cohomology over Laurent series fields, vol. 21 of Algebra and Applications, Springer, [Cham], 2016

    2016

  38. [45]

    M ATHEW, N

    A. M ATHEW, N. N AUMANN , AND J. N OEL, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math., 305 (2017), pp. 994–1084

    2017

  39. [46]

    M OKRANE , La suite spectrale des poids en cohomologie de Hyodo-Kato , Duke Math

    A. M OKRANE , La suite spectrale des poids en cohomologie de Hyodo-Kato , Duke Math. J., 72 (1993), pp. 301–337

    1993

  40. [47]

    M ONSKY AND G

    P. M ONSKY AND G. WASHNITZER , Formal cohomology. I, Ann. of Math. (2), 88 (1968), pp. 181–217

    1968

  41. [48]

    D. R. M ORRISON , The Clemens-Schmid exact sequence and applications , in Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), vol. 106 of Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 1984, pp. 101–119

    1981

  42. [49]

    NAKKAJIMA , Signs in weight spectral sequences, monodromy-weight conjectures, log Hodge symmetry and degenerations of surfaces, Rend

    Y. NAKKAJIMA , Signs in weight spectral sequences, monodromy-weight conjectures, log Hodge symmetry and degenerations of surfaces, Rend. Sem. Mat. Univ. Padova, 116 (2006), pp. 71–185

    2006

  43. [50]

    P ARK, Log motivic nearby cycles

    D. P ARK, Log motivic nearby cycles. https://arxiv.org/abs/2405.14083, 2024

    2024 arXiv

  44. [51]

    R APOPORT AND T

    M. R APOPORT AND T. ZINK, ¨uber die lokale Zetafunktion von Shimuravariet¨aten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math., 68 (1982), pp. 21–101

    1982

  45. [52]

    S AITO , Modules de Hodge polarisables, Publ

    M. S AITO , Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., 24 (1988), pp. 849–995 (1989)

    1988

  46. [53]

    SCHMID , Variation of Hodge structure: the singularities of the period mapping, Invent

    W. SCHMID , Variation of Hodge structure: the singularities of the period mapping, Invent. Math., 22 (1973), pp. 211–319

    1973

  47. [54]

    S CHOLZE , Perfectoid spaces, Publ

    P. S CHOLZE , Perfectoid spaces, Publ. Math. Inst. Hautes ´Etudes Sci., 116 (2012), pp. 245–313

    2012

  48. [55]

    SHIHO , Relative log convergent cohomology and relative rigid cohomology III

    A. SHIHO , Relative log convergent cohomology and relative rigid cohomology III. arXiv:0805.3229 [math.NT], 2008

    2008 arXiv

  49. [56]

    S TEENBRINK , Limits of Hodge structures, Invent

    J. S TEENBRINK , Limits of Hodge structures, Invent. Math., 31 (1975/76), pp. 229–257

    1975

  50. [57]

    T SUZUKI , On base change theorem and coherence in rigid cohomology, Documenta Mathematica, Extra V olume (2003), pp

    N. T SUZUKI , On base change theorem and coherence in rigid cohomology, Documenta Mathematica, Extra V olume (2003), pp. 891–918. in: A Collection of Manuscripts Written in Honour of Kazuya Kato on the Occasion of His Fiftieth Birthday

    2003

  51. [58]

    T UBACH , On the Nori and Hodge realisations of Voevodsky ´etale motives

    S. T UBACH , On the Nori and Hodge realisations of Voevodsky ´etale motives. arXiv:2309.11999 [math.AG], 2024

    2024

  52. [60]

    Math., 155 (2019), pp

    , A motivic version of the theorem of Fontaine and Wintenberger, Compos. Math., 155 (2019), pp. 38–88

    2019

  53. [61]

    Pure Appl

    , Rigid cohomology via the tilting equivalence, J. Pure Appl. Algebra, 223 (2019), pp. 818–843

    2019

  54. [62]

    , The relative (de-)perfectoidification functor and motivic p-adic cohomologies, in Perfectoid spaces, Infosys Sci. Found. Ser., Springer, Singapore, [2022] ©2022, pp. 15–36. 27

    2022

  55. [63]

    , A motivic proof of the finiteness of the relative de Rham cohomology, Arch. Math. (Basel), 123 (2024), pp. 393–398

    2024

  56. [64]

    F. E NRIQUES

    V. VOEVODSKY , Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not., (2002), pp. 351–355. DIPARTIMENTO DI MATEMATICA “F. E NRIQUES ” - U NIVERSIT `A DEGLI STUDI DI MILANO (I TALY) Email address: federico.binda@unimi.it DIPA...

    2002

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.