# 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?

• There is something in this direction in the paper by Deligne-Illusie: see eq. (2.2.1) p. 254 in their paper in Invent. Math. 89. Commented Feb 27, 2014 at 8:49

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."

• To put a condition on Frobenius lift is too strong. For instance, it forces ordinarity. Also I think that the result you are referring to is basically already in Th. 8.8 of the notes of Berthelot-Ogus on crystalline cohomology. Commented Mar 4, 2014 at 8:52
• I don't see how to deduce this from Theorem 8.8 in Berthelot-Ogus, which doesn't mention a lift to $W_{n+1}$ at all. This theorem in degree $\leq p$ is due to Ogus, as I observe. I'm also confused by your ordinarity comment; Frobenius on a supersingular elliptic curve lifts, by Deuring. Are you sure you're not thinking of Frobenius splitting? Commented Mar 4, 2014 at 9:07
• That said, I agree that the condition of a Frobenius lift is strong, as I note in my answer. Commented Mar 4, 2014 at 9:08
• In Berthelot-Ogus, it is supposed that the Frobenius lifts to an entire formal lifting, which is a stronger hypothesis. Nevertheless, the proof of Th. 8.8 in Berthelot-Ogus does not depend on the whole strength of this hypothesis, I think (that is why I wrote 'essentially'). As far as supersingular elliptic curves are concerned, the result of Deuring shows that the curve lifts to a CM curve and therefore a certain power of Frobenius lifts (if $k$ is a finite field). Nevertheless, the lifting will in general be defined over a ramified extension, unlike here. ... Commented Mar 4, 2014 at 17:13
• ... But if you require the Frobenius itself to lift to $W_2(k)$, then the curve has to be ordinary. See for instance Prop. 8.6 in 'Frobenius et dégénérescence de Hodge' by L. Illusie (this is also a result of Nakkajima). Commented Mar 4, 2014 at 17:14