The stacky approach to de Rham-Witt cohomology
Abstract
Let k be a perfect field of characteristic p>0. Ekedahl proved an equivalence of triangulated categories between bounded coherent complexes over the Raynaud ring and complexes of F-gauges. We give a geometric proof, independent of his proof, that the de Rham-Witt realization is a symmetric monoidal equivalence of stable ∞-categories between perfect F-gauges and bounded coherent complexes over the Raynaud ring with Ekedahl's completed star product, as expected by Illusie. The geometry comes from topological restriction homology: TRr(k;Zp)hS1 is an iterated fibre product of copies of TC-(k;Zp), and taking even stacks before and after the fibre product gives the genuine and the naive r-Nygaard stacks. Pushforward from the genuine to the naive r-Nygaard stack realizes Andriopoulos' r-Nygaard filtrations, which interpolate between the Nygaard filtration at r=1 and the Hodge-Witt filtration at r=∞. On the genuine r-Nygaard stack, which is a chain of Nygaard charts, restriction and Frobenius come from the inclusions of sub-chains, Verschiebung from duality, and the differential as a Bockstein operator, and the relations of the star product become projection formulas. As applications, we deduce Ekedahl's Künneth formula and Poincaré duality for de Rham-Witt cohomology from their counterparts for F-gauge cohomology, and show that the special-value formulas of Illusie and of Mondal over finite fields are equivalent.
Related
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BibTeX
@misc{zhang2026stacky,
title = {{The stacky approach to de Rham-Witt cohomology}},
author = {Yuanning Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28149},
eprint = {2610.28149},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28149},
}
RIS
TY - UNPB
TI - The stacky approach to de Rham-Witt cohomology
AU - Yuanning Zhang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28149
UR - https://arxiv.org/abs/2610.28149
AB - Let k be a perfect field of characteristic p>0. Ekedahl proved an equivalence of triangulated categories between bounded coherent complexes over the Raynaud ring and complexes of F-gauges. We give a geometric proof, independent of his proof, that the de Rham-Witt realization is a symmetric monoidal equivalence of stable ∞-categories between perfect F-gauges and bounded coherent complexes over the Raynaud ring with Ekedahl's completed star product, as expected by Illusie. The geometry comes from topological restriction homology: TR^r(k;Z_p)^hS^1 is an iterated fibre product of copies of TC^-(k;Z_p), and taking even stacks before and after the fibre product gives the genuine and the naive r-Nygaard stacks. Pushforward from the genuine to the naive r-Nygaard stack realizes Andriopoulos' r-Nygaard filtrations, which interpolate between the Nygaard filtration at r=1 and the Hodge-Witt filtration at r=∞. On the genuine r-Nygaard stack, which is a chain of Nygaard charts, restriction and Frobenius come from the inclusions of sub-chains, Verschiebung from duality, and the differential as a Bockstein operator, and the relations of the star product become projection formulas. As applications, we deduce Ekedahl's Künneth formula and Poincaré duality for de Rham-Witt cohomology from their counterparts for F-gauge cohomology, and show that the special-value formulas of Illusie and of Mondal over finite fields are equivalent.
ER -