The stacky approach to de Rham-Witt cohomology

Source 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 $\infty$-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: $\mathrm{TR}^r(k;\mathbf Z_p)^{hS^1}$ is an iterated fibre product of copies of $\mathrm{TC}^-(k;\mathbf Z_p)$, 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 $r=\infty$. 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.