Decomposition of de Rham complex for quasi-F-split varieties
Alexander Petrov
TL;DR
The paper resolves long-standing questions about de Rham cohomology in positive characteristic by proving that smooth quasi-$F$-split varieties admit a Frobenius-pullback decomposition of their de Rham complex, $F_{X/k*}\Omega^{\bullet}_{X/k} \simeq \bigoplus_{i\ge0} \Omega^i_{X^{(1)}/k}[-i]$, which implies $E_1$-degeneration of the Hodge-to-de Rham spectral sequence and Kodaira–Nakano vanishing for smooth proper cases. The central tool is the de Rham stack, whose gerbe structure yields a clean decomposition after Frobenius pullback, and which also facilitates extending the decomposability to smooth Artin stacks, including proving Frobenius splitting of classifying stacks $BG$ for reductive groups. An alternative semi-perfect descent approach provides a second route to the same decomposition, underlining the robustness of the Frobenius-based splitting phenomenon. Together, these results broaden the scope of degeneracy and vanishing theorems in characteristic $p$, connect de Rham cohomology with stack-theoretic methods, and yield concrete consequences for the cohomology of classifying stacks and related geometric objects.
Abstract
Using the de Rham stack of Bhatt-Lurie and Drinfeld, we prove that de Rham complex of a smooth quasi-F-split variety over a perfect field of positive characteristic decomposes in all degrees. In particular, smooth proper quasi-F-split varieties have degenerate Hodge-to-de Rham spectral sequence, and satisfy Kodaira-Akizuki-Nakano vanishing. We apply this to prove that the Hodge-to-de Rham spectral sequence for the classifying stack of a reductive group over a field of positive characteristic degenerates.