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On de Rham--Witt Cohomology of Classifying Stacks

T0 review · reviewed 2026-05-13 · grok-4.3

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Pith's one-line read A proper smooth fourfold over a perfect field of characteristic p>0 is constructed with asymmetric Hodge-Witt numbers in total degree 3 via computation of the Hodge-Witt cohomology of the classifying stack B alpha_p.

arxiv 2604.03062 v1 submitted 2026-04-03 math.AG math.NT

Shizhang Li , Yuan Yang This is my paper · ORCID
classification math.AGmath.NT
keywords exampleclassifyingcohomologydegreehodge--witttotalalphaapproximating
verification ladder T0 review T1 audit T2 compute T3 formal

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Hodge numbers count independent cycles or 'holes' in different dimensions on geometric objects like surfaces or higher-dimensional varieties. In complex geometry these counts are symmetric. In positive characteristic a refined version called Hodge-Witt numbers arises from de Rham-Witt cohomology, which incorporates Witt vector rings to handle p-power phenomena. The paper produces a concrete four-dimensional smooth proper variety over a perfect field of characteristic p where the Hodge-Witt numbers in total degree 3 fail to be symmetric. The construction proceeds by computing and approximating the relevant cohomology groups on the classifying stack B alpha_p, a stack that classifies alpha_p-torsors. Alpha_p is the kernel of the Frobenius endomorphism on the additive group in characteristic p. This yields the first known example at this low dimension and degree, showing that the symmetry familiar from characteristic zero does not hold in general for Hodge-Witt numbers.
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Core claim

We give an example of proper smooth fourfold over a perfect field k of characteristic p > 0 with asymmetric Hodge--Witt numbers in total degree 3. Our example is sharp both in terms of dimension and total degree.

Load-bearing premise

The computation and approximation of the Hodge-Witt cohomology groups of the classifying stack B alpha_p produce a valid proper smooth fourfold exhibiting the claimed asymmetry in degree 3.

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Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on standard properties of de Rham-Witt cohomology for smooth schemes and stacks over perfect fields of positive characteristic together with the existence of the classifying stack B alpha_p; no free parameters or new entities are introduced.

assumptions (1)
  • standard math De Rham-Witt cohomology is well-defined and functorial for smooth schemes and algebraic stacks over perfect fields of characteristic p
    This is a standard background result in the theory of crystalline and Witt cohomology.

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Pith. "Pith review of On de Rham--Witt Cohomology of Classifying Stacks." pith.science (2026). https://pith.science/paper/2604.03062

@misc{pith2026260403062,
  author       = {Pith},
  title        = {Pith review of: On de Rham--Witt Cohomology of Classifying Stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.03062}},
  note         = {Machine review of arXiv:2604.03062}
}
read the original abstract

We give an example of proper smooth fourfold over a perfect field k of characteristic p > 0 with asymmetric Hodge--Witt numbers in total degree 3. Our example is sharp both in terms of dimension and total degree. We arrive at our example by computing and approximating the Hodge--Witt cohomology groups of the classifying stack B alpha_p.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Higher dimensional dominoes in de Rham-Witt cohomology

    math.AG 2026-07 accept novelty 7.0 of 10

    Two-dimensional dominoes in de Rham–Witt cohomology are classified by orbits of Frobenius-skew polynomials, all dominoes admit a triangular normal form, and the resulting theory bounds the p-primary Brauer exponent of...

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Works this paper leans on

4 extracted references · 4 canonical work pages · cited by 1 Pith paper

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