In a beautiful paper Deligne and Illusie have shown the following: Let $f\colon X \to S$ be a smooth proper morphism of schemes in characteristic $p > 0$, let $F\colon X \to X^{(p)}$ be the relative Frobenius and let $b$ be an integer. Assume that $\tau_{\leq b}F_*\Omega_{X/S}$ decomposes in the derived category of coherent sheaves on $X^{(p)}$ (where $\Omega_{X/S}$ denotes the De-Rham complex). Then for $i+j \leq b$ one has:

(1) The sheaves $R^if_*\Omega^j_{X/S}$ are locally free and their formation commutes with arbitrary base change.

(2) The Hodge spectral sequence for $f$ satisfies $E^{ij}_1 = E^{ij}_{\infty}$.

(see *Relevements modulo $p^2$ et decomposition du complexe de de Rham*, Inv. Math. 89).
As also explained in that paper, the assumption is satisfied for $b = p-1$ and if $X^{(p)}$ can be lifted over a $\tilde S$, where $\tilde S$ is a flat $\mathbb Z/p^2\mathbb Z$-schemes whose reduction modulo $p$ is $S$. The result then can be applied to show the degeneration of the Hodge spectral sequence and the vanishing result of Kodaira-Akizuki-Nakano in characteristic $0$ (see also H. Esnault; E. Viehweg: *Lectures on vanishing theorems*, Birkhäuser 1992).

**Question**: Does there exist a reference for a similar result for a morphism of algebraic stacks?

It seems to me that using Chow's Lemma for stacks by Olsson the proof should be completely analogous as the proof of Deligne and Illusie (at least for tame Deligne-Mumford stacks), but a quick browsing did not produce any results except for a paper of Matsuki and Olsson (*Kawamata-Viehweg vanishing as Kodaira vanishing for stacks*, Math. Res. Letters 12 (2005)), where the above result of Kodaira-Akizuki-Nakano is generalized. There the authors claim that the vanishing result can be proved by the same arguments as in the paper of Deligne and Illusie. I do not doubt this claim but as a reference this is not particularly helpful.

**Remark**: Kato has also sketched the proof of a generalisation of Deligne's and Illusie's result to arbitrary log schemes. This should of course be a special case of a generalization to algebraic stacks via the work of Olsson.