Degeneration Theorems of Connes and Feigin--Tsygan Type in Mixed Characteristic, with q-Analogues

We prove mixed-characteristic analogues of the Connes and Feigin–Tsygan degeneration theorem. Let $W=W(k)$ be the Witt vectors of a perfect field of characteristic $p>0$. For a smooth proper variety $X$ over $W$, the de Rham-to-$\text{HP}$ spectral sequence is split degenerate under the small-dimension hypothesis $dim(X\mathrm{/}W)<p-1$. More generally, if $X$ is smooth and proper over the ring of integers $O_K$ of a finite extension of $\mathrm{F}\mathrm{r}\mathrm{a}\mathrm{c}(W)$ with ramification index $e$, we prove the corresponding split degeneration under $2edim(X\mathrm{/}{O}_K)<p-1$. Under the same ramification hypothesis, we also prove split degeneration of the $Ainf$-to-$TP$ spectral sequence. Finally, after inverting an explicit factorial, we obtain a topological $q$-de Rham analogue.