Hodge--Tate splitting and Akizuki--Nakano vanishing in positive characteristic
We introduce the notion of Hodge--Tate splitting for schemes in positive characteristic. For a smooth variety $X$ over a perfect field $k$ of characteristic $p > 0$, we say that $X$ is Hodge--Tate split if the natural morphism $\mathcal{O}_X \to F_*\Omega^\bullet_{X/k}$ induced by the absolute Frobenius admits a splitting in $\mathcal{D}_{\mathrm{qcoh}}(X)$. We prove that this condition is equivalent to the existence of a decomposition $F_*\Omega^\bullet_{X/k} \simeq \bigoplus_{i=0}^{\dim X}\Omega^i_{X/k}[-i]$ of its de Rham complex. Consequently, for smooth projective varieties, Hodge--Tate splitting implies Akizuki--Nakano vanishing and the $E_1$-degeneration of the Hodge-to-de Rham spectral sequence. Furthermore, we establish criteria and permanence properties for Hodge--Tate splitting and use them to construct many new examples of varieties whose de Rham complexes decompose. These include blow-ups of quasi-$F$-split varieties along strata of simple normal crossings divisors, complete intersections in toric varieties, and linearly reductive quotients of Hodge--Tate split varieties. Among these examples, we obtain smooth projective varieties whose Hodge-to-de Rham spectral sequences degenerate at $E_1$, whereas their Hochschild--Kostant--Rosenberg spectral sequences do not degenerate. As an application in mixed characteristic, we prove an Akizuki--Nakano-type vanishing theorem for smooth projective globally $+$-regular varieties over the Witt ring of a perfect field.