Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring? Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:

The Hodge-De Rham spectral sequence $E^{a,b}_1=R^af∗Ω^b(X/S)⇒H^{a+b}_{DR}(X/S)$ degenerates in $E_1$.

This is also true when $S=Spec(k)$ with $k$ a perfect field of characteristic $p$, and $X$ has a smooth $W_2(k)$-lifing. Here $W_2(k)$ is the length 2 truncated Witt ring with values in $k$.  
My question is when $S= Spec(W_2(k))$, and $X$ has a smooth $W_3(k)$-lifting, does the statement of $E_1$ degeneration still hold (maybe one needs more conditions)? Has anyone worked on this kind of question before? 
 A: First of all, the statement of Deligne-Illusie as you've given it is slightly wrong; there is only degeneration in degree $\leq \dim(X)$ (and thus full degeneration if $\dim(X)\leq p$).  
Yukiyoshi Nakkajima proves an analogous result of the sort you'd like (extending work of Ogus) under the condition that the smooth proper scheme $X/W_n(k), n\geq 2$ admits a Frobenius lift.  Here is the statement:

Theorem. Let $X$ be a smooth proper scheme over $W_n(k)$ ($k$ perfect of characteristic $p$, $n\geq 2$).  If $X$ admits a lift of the Frobenius of $X\otimes_{W_n(k)} k$ and has a smooth lifting $\tilde{X}$ over $W_{n+1}(k)$,
  then the Hodge-de Rahm spectral sequence
  $$E^{i,j}_1=H^j(X, \Omega^i_{X/W_n(k)})\implies H^{i+j}_{dR}(X/W_n(k))$$
  degenerates at $E_1$.

Note that unlike the Deligne-Illusie result, there is no condition on the dimension of $X$ or the degree of the Hodge cohomology; rather the whole spectral sequence degenerates.  On the other hand, this result requires the (strong) condition of a Frobenius lift, which is vacuous for Deligne-Illusie (where $n=1$).
Ogus proves a similar result (with a degree condition) in 8.2.6 of "F-crystals, Griffiths Transversality, and the Hodge Decomposition."
