The Stacks project

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

\[ E_1^{p, q} = H^ q(X, \Omega ^ p_{X/S}) \]

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

\[ \xymatrix{ H^2(X, \mathcal{O}_ X) \ar[r] \ar@{-->}[rrd] \ar@{..>}[rrrdd] & H^2(X, \Omega ^1_{X/S}) \ar[r] \ar@{-->}[rrd] & H^2(X, \Omega ^2_{X/S}) \ar[r] & H^2(X, \Omega ^3_{X/S}) \\ H^1(X, \mathcal{O}_ X) \ar[r] \ar@{-->}[rrd] & H^1(X, \Omega ^1_{X/S}) \ar[r] \ar@{-->}[rrd] & H^1(X, \Omega ^2_{X/S}) \ar[r] & H^1(X, \Omega ^3_{X/S}) \\ H^0(X, \mathcal{O}_ X) \ar[r] & H^0(X, \Omega ^1_{X/S}) \ar[r] & H^0(X, \Omega ^2_{X/S}) \ar[r] & H^0(X, \Omega ^3_{X/S}) } \]

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

\[ H^0_{dR}(X/S) = \mathop{\mathrm{Ker}}(\text{d} : H^0(X, \mathcal{O}_ X) \to H^0(X, \Omega ^1_{X/S})) \]

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

\[ E_2^{p, q} = H^ p(X, H^ q(\Omega ^\bullet _{X/S})) = H^ p(X, \mathcal{H}^ q) \]

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

\[ \xymatrix{ H^0(X, \mathcal{H}^2) \ar[rrd] \ar@{..>}[rrrdd] & H^1(X, \mathcal{H}^2) \ar[rrd] & H^2(X, \mathcal{H}^2) & H^3(X, \mathcal{H}^2) \\ H^0(X, \mathcal{H}^1) \ar[rrd] & H^1(X, \mathcal{H}^1) \ar[rrd] & H^2(X, \mathcal{H}^1) & H^3(X, \mathcal{H}^1) \\ H^0(X, \mathcal{H}^0) & H^1(X, \mathcal{H}^0) & H^2(X, \mathcal{H}^0) & H^3(X, \mathcal{H}^0) } \]

Looking in degree $0$ we conclude that

\[ H^0_{dR}(X/S) = H^0(X, \mathcal{H}^0) \]

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

\[ 0 \to H^1(X, \mathcal{H}^0) \to H^1_{dR}(X/S) \to H^0(X, \mathcal{H}^1) \to H^2(X, \mathcal{H}^0) \to H^2_{dR}(X/S) \]

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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