On de Rham--Witt Cohomology of Classifying Stacks

Asymmetry in Hodge--Witt Numbers for Classifying Stacks: The Case of $B\alpha_p$

Introduction and Context

This paper investigates the structure of de Rham--Witt cohomology in positive characteristic, focusing on the Hodge--Witt numbers associated to smooth proper varieties. These are $p$-adic invariants introduced by Ekedahl as characteristic $p$ analogues of the classical Hodge numbers, capturing subtle information about the slope filtration and torsion phenomena in crystalline cohomology. The central question is whether the symmetry $h_W^{i,j} = h_W^{j,i}$, which holds for the classical Hodge numbers in characteristic zero and persists in low degrees or dimensions in positive characteristic, always holds for Hodge--Witt numbers.

The authors produce—by explicit calculation—a smooth proper fourfold over a perfect field of characteristic $p > 0$ whose Hodge--Witt numbers are asymmetric in total degree $3$, thereby establishing the sharp threshold for the failure of Hodge symmetry in this context. The construction is rooted in an analysis of the de Rham--Witt cohomology of the classifying stack $B\alpha_p$, where $\alpha_p$ is the height one local-local finite group scheme.

De Rham--Witt Cohomology and Hodge--Witt Numbers

The de Rham--Witt complex $W\Omega_{X/k}^\bullet$ is a filtered, graded complex equipped with Frobenius and Verschiebung operators and a canonical differential, serving as a canonical model for crystalline cohomology for smooth $k$-schemes. For smooth proper schemes $p$0, its hypercohomology defines the main object $p$1. The homological algebra of this object leads to invariants including slope numbers, domino numbers, and Hodge--Witt numbers $p$2, defined via a mixture of slope multiplicities and torsion invariants (domino numbers).

There are structural results: Ekedahl proved that $p$3 always satisfy Serre duality, and Hodge symmetry holds for $p$4 or $p$5. This leaves open the question in higher dimensions and degrees.

De Rham--Witt Cohomology for Stacks

The authors extend the construction of de Rham--Witt cohomology to geometric Artin stacks and show that, under a Hodge-properness condition (strong finiteness for Hodge cohomology), the resulting complexes remain coherent in the sense of Illusie--Raynaud. This provides a natural framework for considering de Rham--Witt invariants of stacks such as classifying stacks $p$6 for finite, flat commutative group schemes $p$7.

They construct a natural filtered complex and develop a spectral sequence—based on the diagonal $p$8-structure introduced by Ekedahl—for computing associated graded pieces of the de Rham--Witt cohomology for stacks built as simplicial schemes, as is the case for $p$9.

Calculation for $p$0 and Smooth Proper Fourfolds

The key computational content comes from analyzing the classifying stack $p$1. By realizing $p$2 as a smooth stack with a presentation by an $p$3-torsor abelian variety, the authors obtain a simplicial scheme whose terms involve products of abelian varieties. They filter the associated de Rham--Witt complexes via the diagonal $p$4-structure and compute the first nontrivial differentials.

The main result is:

There exists a smooth proper $p$5-fold $p$6 over a perfect field of characteristic $p$7 with $p$8, $p$9, $h_W^{i,j} = h_W^{j,i}$0, and $h_W^{i,j} = h_W^{j,i}$1.

The calculation is explicit. The authors first show, via a spectral sequence, that the potential symmetry $h_W^{i,j} = h_W^{j,i}$2 fails, essentially because of nonsplit extensions by domino modules (elementary objects capturing the 'jump' in the filtration) and the structure of the derived category of graded $h_W^{i,j} = h_W^{j,i}$3-modules. The crucial step is the identification of the $h_W^{i,j} = h_W^{j,i}$4-term $h_W^{i,j} = h_W^{j,i}$5 of their spectral sequence with a non-split extension, producing the asymmetric domino module $h_W^{i,j} = h_W^{j,i}$6.

They realize this failure geometrically by invoking a result due to Antieau--Bhatt--Mathew: for any finite $h_W^{i,j} = h_W^{j,i}$7-group scheme $h_W^{i,j} = h_W^{j,i}$8, smooth projective varieties of arbitrarily large dimension can be constructed admitting maps to $h_W^{i,j} = h_W^{j,i}$9 that are de Rham--Witt equivalences up to large degrees. In particular, in dimension $p > 0$0, such varieties approximate $p > 0$1 in low degrees, preserving the Hodge--Witt structure causing asymmetry.

Implications and Theoretical Developments

This work provides a sharp bound for the failure of Hodge symmetry for Hodge--Witt numbers, paralleling the classical phenomena for Hodge numbers in characteristic $p > 0$2. The techniques—mixing modern $p > 0$3-categorical language, stack cohomology, and explicit spectral sequence analysis—connect the behavior of stacky invariants to the structure of coherent graded modules and derived functors.

The result shows that the familiar symmetries from complex geometry break down in precisely computable ways in characteristic $p > 0$4, and that stack-theoretic methods are indispensable for constructing and understanding smooth projective varieties with these pathologies.

Practically, these computations may impact the study of slope and Newton polygons, the structure of supersingular varieties, and arithmetic questions in crystalline cohomology. The machinery developed here (e.g., $p > 0$5-categorical star products on de Rham--Witt complexes, explicit resolution of stacks, coherent module characterizations) is likely of independent utility for questions about $p > 0$6-adic Hodge theory, prismatic cohomology, and deformation theory of stacks.

Future work may push these methods to broader classes of stacks, higher genus, or seek finer invariants distinguishing non-symmetric scenarios, or connect with prismatic and syntomic cohomology frameworks.

Conclusion

This paper establishes that Hodge--Witt numbers for smooth proper varieties are not symmetric in general—demonstrated by direct calculation for a fourfold approximating the classifying stack $p > 0$7—with the asymmetry precisely appearing in total degree $p > 0$8 and dimension $p > 0$9. The implications are both conceptual, refining the landscape of $3$0-adic invariants, and methodological, providing a computational template for further exploration of stacky and filtered cohomological invariants in positive characteristic (2604.03062).