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:



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



Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring?
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](http://ac.els-cdn.com/S0022314X97921120/1-s2.0-S0022314X97921120-main.pdf?_tid=980c604c-a312-11e3-8ad3-00000aab0f27&acdnat=1393878780_e05019a1c126eee41afae5a76da4f91d) 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:



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> **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 $E\_1$.
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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."


