## 50.6 Two spectral sequences

Let $p : X \to S$ be a morphism of schemes. Since the category of $p^{-1}\mathcal{O}_ S$-modules on $X$ has enough injectives there exist a Cartan-Eilenberg resolution for $\Omega ^\bullet _{X/S}$. See Derived Categories, Lemma 13.21.2. Hence we can apply Derived Categories, Lemma 13.21.3 to get two spectral sequences both converging to the de Rham cohomology of $X$ over $S$.

The first is customarily called *the Hodge-to-de Rham spectral sequence*. The first page of this spectral sequence has

which are the Hodge cohomology groups of $X/S$ (whence the name). The differential $d_1$ on this page is given by the maps $d_1^{p, q} : H^ q(X, \Omega ^ p_{X/S}) \to H^ q(X. \Omega ^{p + 1}_{X/S})$ induced by the differential $\text{d} : \Omega ^ p_{X/S} \to \Omega ^{p + 1}_{X/S}$. Here is a picture

where we have drawn striped arrows to indicate the source and target of the differentials on the $E_2$ page and a dotted arrow for a differential on the $E_3$ page. Looking in degree $0$ we conclude that

Of course, this is also immediately clear from the fact that the de Rham complex starts in degree $0$ with $\mathcal{O}_ X \to \Omega ^1_{X/S}$.

The second spectral sequence is usually called *the conjugate spectral sequence*. The second page of this spectral sequence has

where $\mathcal{H}^ q = H^ q(\Omega ^\bullet _{X/S})$ is the $q$th cohomology sheaf of the de Rham complex of $X/S$. The differentials on this page are given by $E_2^{p, q} \to E_2^{p + 2, q - 1}$. Here is a picture

Looking in degree $0$ we conclude that

which is obvious if you think about it. In degree $1$ we get an exact sequence

It turns out that if $X \to S$ is smooth and $S$ lives in characteristic $p$, then the sheaves $\mathcal{H}^ q$ are computable (in terms of a certain sheaves of differentials) and the conjugate spectral sequence is a valuable tool (insert future reference here).

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