pith. sign in

arxiv: 2606.05260 · v2 · pith:77DDA5Y6new · submitted 2026-06-03 · 🧮 math.AG · math.AC

A Counterexample to Bhatt-Lurie's Cohomological Dimension Conjecture

Guo Li This is my paper

Pith reviewed 2026-06-28 04:09 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords Bhatt-Lurie conjecturecounterexamplecohomological dimensionHodge-Tate locusregular local ringsnon-excellent ringsdiscrete valuation ring
0
0 comments X

The pith

A non-excellent discrete valuation ring provides a counterexample to Bhatt-Lurie's conjecture on the cohomological dimension of the Hodge-Tate locus for regular local rings.

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

Bhatt and Lurie conjectured an upper bound on the cohomological dimension of the Hodge-Tate locus when the base ring is regular. The paper exhibits an explicit counterexample using a non-excellent discrete valuation ring constructed by Datta and Smith. This ring satisfies the regularity hypotheses required for the Hodge-Tate locus to be defined, yet its cohomological dimension exceeds the conjectured bound. The paper further shows that the same construction produces broader families of counterexamples and that the expected bound holds once an excellence hypothesis is imposed.

Core claim

We exhibit a counterexample to a conjecture of Bhatt--Lurie on the cohomological dimension of the Hodge--Tate locus for regular local rings. The example arises from a non-excellent discrete valuation ring constructed by Datta--Smith, closely related to an earlier example of Bosch--Lütkebohmert--Raynaud. We also explain how the same mechanism yields broader families of counterexamples, while the expected bound is recovered under an excellence hypothesis.

What carries the argument

The non-excellent discrete valuation ring of Datta-Smith, which meets regularity conditions for the Hodge-Tate locus while allowing its cohomological dimension to exceed the conjectured bound.

Load-bearing premise

The Datta-Smith non-excellent discrete valuation ring satisfies the regularity and local ring hypotheses needed for the Hodge-Tate locus to be defined while violating the conjectured cohomological dimension bound.

What would settle it

A direct computation establishing that the cohomological dimension of the Hodge-Tate locus on the Datta-Smith ring stays within the conjectured bound, or a proof that this ring fails to be regular.

read the original abstract

We exhibit a counterexample to a conjecture of Bhatt--Lurie on the cohomological dimension of the Hodge--Tate locus for regular local rings. The example arises from a non-excellent discrete valuation ring constructed by Datta--Smith, closely related to an earlier example of Bosch--L\"{u}tkebohmert--Raynaud. We also explain how the same mechanism yields broader families of counterexamples, while the expected bound is recovered under an excellence hypothesis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper exhibits a counterexample to the Bhatt-Lurie conjecture on the cohomological dimension of the Hodge-Tate locus for regular local rings. The counterexample is constructed from a non-excellent discrete valuation ring due to Datta-Smith (closely related to an earlier Bosch-Lütkebohmert-Raynaud example), and the authors recover the conjectured bound under an excellence hypothesis while also producing broader families of counterexamples.

Significance. If the verification holds, the result is significant: it supplies a concrete counterexample to a conjecture in p-adic Hodge theory / algebraic geometry, isolates excellence as the key hypothesis needed for the bound, and demonstrates that the failure is not an artifact of pathology but arises from a standard (if non-excellent) construction. The explicit recovery of the bound under excellence is a positive contribution that clarifies the conjecture's natural scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; counterexample relies on external construction

full rationale

The paper's central claim is the existence of a counterexample to the Bhatt-Lurie conjecture, constructed from the non-excellent DVR of Datta-Smith (an external reference). No load-bearing step reduces to a self-definition, fitted parameter renamed as prediction, or self-citation chain; the argument verifies that this ring satisfies the stated regularity hypotheses while violating the bound, and recovers the bound under excellence as a consistency check. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard facts about discrete valuation rings, regularity, and the definition of the Hodge-Tate locus; no free parameters or invented entities are introduced.

axioms (1)
  • standard math Standard properties of cohomology theories and regular local rings in algebraic geometry hold for the Datta-Smith construction.
    Invoked implicitly when applying the counterexample to the conjecture.

pith-pipeline@v0.9.1-grok · 5594 in / 1143 out tokens · 38471 ms · 2026-06-28T04:09:54.113682+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 4 canonical work pages

  1. [1]

    v-vector bundles on p -adic fields and

    Ansch. v-vector bundles on p -adic fields and. 2025 , eprint=

    2025

  2. [2]

    Forum of Mathematics, Pi , year=

    Johannes Ansch. Forum of Mathematics, Pi , year=

  3. [3]

    2026 , urldate =

    Alper, Jarod , title =. 2026 , urldate =

    2026

  4. [4]

    2023 , eprint=

    Cartier smoothness in prismatic cohomology , author=. 2023 , eprint=

    2023

  5. [5]

    2018 , note =

    Bhatt, Bhargav , title =. 2018 , note =

    2018

  6. [6]

    2022 , eprint =

    Absolute prismatic cohomology , author =. 2022 , eprint =

    2022

  7. [7]

    2022 , eprint =

    The prismatization of p -adic formal schemes , author =. 2022 , eprint =

    2022

  8. [8]

    Bosch, Siegfried and L. N. 1990 , doi =

    1990

  9. [9]

    Syntomic complexes and p -adic étale Tate twists , volume=

    Bhatt, Bhargav and Mathew, Akhil , year=. Syntomic complexes and p -adic étale Tate twists , volume=. doi:10.1017/fmp.2022.21 , journal=

    work page doi:10.1017/fmp.2022.21 2022

  10. [10]

    Integral p-adic

    Bhargav Bhatt and Matthew Morrow and Peter Scholze , year=. Integral p-adic. 1602.03148 , archivePrefix=

    Pith/arXiv arXiv

  11. [11]

    Topological

    Bhargav Bhatt and Matthew Morrow and Peter Scholze , year=. Topological. 1802.03261 , archivePrefix=

    Pith/arXiv arXiv

  12. [12]

    Annals of Mathematics , volume =

    Prisms and prismatic cohomology , author =. Annals of Mathematics , volume =. 2022 , publisher =. doi:10.4007/annals.2022.196.3.5 , eprint =

    work page doi:10.4007/annals.2022.196.3.5 2022

  13. [13]

    Prismatic F -crystals and crystalline G alois representations

    Bhatt, Bhargav and Scholze, Peter , journal =. Prismatic F -crystals and crystalline G alois representations. 2023 , doi =. 2106.14735 , archivePrefix =

    arXiv 2023

  14. [14]

    2018 , eprint=

    Excellence in prime characteristic , author=. 2018 , eprint=

    2018

  15. [15]

    2024 , eprint=

    Prismatization , author=. 2024 , eprint=

    2024

  16. [16]

    A prismatic approach to ( , G ) -modules and F -crystals

    Du, Heng and Liu, Tong , journal =. A prismatic approach to ( , G ) -modules and F -crystals. 2023 , note =. doi:10.4171/JEMS , eprint =

    work page doi:10.4171/jems 2023

  17. [17]

    2311.07024 , archivePrefix=

    Hui Gao and Yu Min and Yupeng Wang , year=. 2311.07024 , archivePrefix=

    arXiv

  18. [18]

    Prismatic crystals over the de

    Hui Gao and Yu Min and Yupeng Wang , year=. Prismatic crystals over the de. 2206.10276 , archivePrefix=

    arXiv

  19. [19]

    Min, Yu and Wang, Yupeng , title =. J. Eur. Math. Soc. , year =. doi:10.4171/JEMS/1579 , url =

    work page doi:10.4171/jems/1579

  20. [20]

    2006 , eprint=

    Andre-Quillen homology of commutative algebras , author=. 2006 , eprint=

    2006

  21. [21]

    Logarithmic

    Teruhisa Koshikawa , year=. Logarithmic. 2007.14037 , archivePrefix=

    arXiv 2007

  22. [22]

    On Noetherian Rings of Characteristic p , urldate =

    Ernst Kunz , journal =. On Noetherian Rings of Characteristic p , urldate =

  23. [23]

    De R ham prismatic crystals over O _K

    Liu, Zeyu , journal =. De R ham prismatic crystals over O _K. 2023 , doi =. 2203.02425 , archivePrefix =

    arXiv 2023

  24. [24]

    On the prismatization of O _K beyond the H odge-- T ate locus

    Liu, Zeyu , year =. On the prismatization of O _K beyond the H odge-- T ate locus. 2409.02051 , archivePrefix =

    arXiv

  25. [25]

    2025 , eprint =

    A stacky approach to prismatic crystals via q -prism charts , author =. 2025 , eprint =

    2025

  26. [26]

    1980 , note =

    Matsumura, Hideyuki , title =. 1980 , note =

    1980

  27. [27]

    2023 , eprint =

    Absolute calculus and prismatic crystals on cyclotomic rings , author =. 2023 , eprint =

    2023

  28. [28]

    2021 , eprint=

    Valuation rings of dimension one as limits of smooth algebras , author=. 2021 , eprint=

    2021

  29. [29]

    Sen, Shankar , journal =

  30. [30]

    The Stacks project , howpublished =

    The. The Stacks project , howpublished =

  31. [31]

    Journal f

    Finiteness and duality for the cohomology of prismatic crystals , author =. Journal f. 2023 , doi =. 2109.00801 , archivePrefix =

    arXiv 2023

  32. [32]

    Prismatic crystals and q -Higgs fields

    Takeshi Tsuji , year=. Prismatic crystals and q -Higgs fields. 2403.11676 , archivePrefix=

    arXiv

  33. [33]

    G alois representations, ( , ) -modules and prismatic F -crystals

    Wu, Zhiyou , journal =. G alois representations, ( , ) -modules and prismatic F -crystals. 2021 , doi =. 2104.12105 , archivePrefix =

    arXiv 2021