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

On de Rham--Witt Cohomology of Classifying Stacks

Shizhang Li , Yuan Yang This is my paper

Pith reviewed 2026-05-13 17:49 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords exampleclassifyingcohomologydegreehodge--witttotalalphaapproximating
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The pith

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.

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

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.

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.

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.

Editorial analysis

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Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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.

pith-pipeline@v0.9.0 · 5339 in / 1349 out tokens · 114741 ms · 2026-05-13T17:49:06.802106+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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