20.13 The Leray spectral sequence
Lemma 20.13.1. Let $f : X \to Y$ be a morphism of ringed spaces. There is a commutative diagram
\[ \xymatrix{ D^{+}(X) \ar[rr]_-{R\Gamma (X, -)} \ar[d]_{Rf_*} & & D^{+}(\mathcal{O}_ X(X)) \ar[d]^{\text{restriction}} \\ D^{+}(Y) \ar[rr]^-{R\Gamma (Y, -)} & & D^{+}(\mathcal{O}_ Y(Y)) } \]
More generally for any $V \subset Y$ open and $U = f^{-1}(V)$ there is a commutative diagram
\[ \xymatrix{ D^{+}(X) \ar[rr]_-{R\Gamma (U, -)} \ar[d]_{Rf_*} & & D^{+}(\mathcal{O}_ X(U)) \ar[d]^{\text{restriction}} \\ D^{+}(Y) \ar[rr]^-{R\Gamma (V, -)} & & D^{+}(\mathcal{O}_ Y(V)) } \]
See also Remark 20.13.2 for more explanation.
Proof.
Let $\Gamma _{res} : \textit{Mod}(\mathcal{O}_ X) \to \text{Mod}_{\mathcal{O}_ Y(Y)}$ be the functor which associates to an $\mathcal{O}_ X$-module $\mathcal{F}$ the global sections of $\mathcal{F}$ viewed as an $\mathcal{O}_ Y(Y)$-module via the map $f^\sharp : \mathcal{O}_ Y(Y) \to \mathcal{O}_ X(X)$. Let $restriction : \text{Mod}_{\mathcal{O}_ X(X)} \to \text{Mod}_{\mathcal{O}_ Y(Y)}$ be the restriction functor induced by $f^\sharp : \mathcal{O}_ Y(Y) \to \mathcal{O}_ X(X)$. Note that $restriction$ is exact so that its right derived functor is computed by simply applying the restriction functor, see Derived Categories, Lemma 13.16.9. It is clear that
\[ \Gamma _{res} = restriction \circ \Gamma (X, -) = \Gamma (Y, -) \circ f_* \]
We claim that Derived Categories, Lemma 13.22.1 applies to both compositions. For the first this is clear by our remarks above. For the second, it follows from Lemma 20.11.10 which implies that injective $\mathcal{O}_ X$-modules are mapped to $\Gamma (Y, -)$-acyclic sheaves on $Y$.
$\square$
Lemma 20.13.3. Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module.
The cohomology groups $H^ i(U, \mathcal{F})$ for $U \subset X$ open of $\mathcal{F}$ computed as an $\mathcal{O}_ X$-module, or computed as an abelian sheaf are identical.
Let $f : X \to Y$ be a morphism of ringed spaces. The higher direct images $R^ if_*\mathcal{F}$ of $\mathcal{F}$ computed as an $\mathcal{O}_ X$-module, or computed as an abelian sheaf are identical.
There are similar statements in the case of bounded below complexes of $\mathcal{O}_ X$-modules.
Proof.
Consider the morphism of ringed spaces $(X, \mathcal{O}_ X) \to (X, \underline{\mathbf{Z}}_ X)$ given by the identity on the underlying topological space and by the unique map of sheaves of rings $\underline{\mathbf{Z}}_ X \to \mathcal{O}_ X$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Denote $\mathcal{F}_{ab}$ the same sheaf seen as an $\underline{\mathbf{Z}}_ X$-module, i.e., seen as a sheaf of abelian groups. Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. By Remark 20.13.2 we see that $\Gamma (X, \mathcal{I}^\bullet )$ computes both $R\Gamma (X, \mathcal{F})$ and $R\Gamma (X, \mathcal{F}_{ab})$. This proves (1).
To prove (2) we use (1) and Lemma 20.7.3. The result follows immediately.
$\square$
Lemma 20.13.4 (Leray spectral sequence). Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_ X$-modules. There is a spectral sequence
\[ E_2^{p, q} = H^ p(Y, R^ qf_*(\mathcal{F}^\bullet )) \]
converging to $H^{p + q}(X, \mathcal{F}^\bullet )$.
Proof.
This is just the Grothendieck spectral sequence Derived Categories, Lemma 13.22.2 coming from the composition of functors $\Gamma _{res} = \Gamma (Y, -) \circ f_*$ where $\Gamma _{res}$ is as in the proof of Lemma 20.13.1. To see that the assumptions of Derived Categories, Lemma 13.22.2 are satisfied, see the proof of Lemma 20.13.1 or Remark 20.13.2.
$\square$
Lemma 20.13.6. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module.
If $R^ qf_*\mathcal{F} = 0$ for $q > 0$, then $H^ p(X, \mathcal{F}) = H^ p(Y, f_*\mathcal{F})$ for all $p$.
If $H^ p(Y, R^ qf_*\mathcal{F}) = 0$ for all $q$ and $p > 0$, then $H^ q(X, \mathcal{F}) = H^0(Y, R^ qf_*\mathcal{F})$ for all $q$.
Proof.
These are two simple conditions that force the Leray spectral sequence to degenerate at $E_2$. You can also prove these facts directly (without using the spectral sequence) which is a good exercise in cohomology of sheaves.
$\square$
slogan
Lemma 20.13.7. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. In this case $Rg_* \circ Rf_* = R(g \circ f)_*$ as functors from $D^{+}(X) \to D^{+}(Z)$.
Proof.
We are going to apply Derived Categories, Lemma 13.22.1. It is clear that $g_* \circ f_* = (g \circ f)_*$, see Sheaves, Lemma 6.21.2. It remains to show that $f_*\mathcal{I}$ is $g_*$-acyclic. This follows from Lemma 20.11.10 and the description of the higher direct images $R^ ig_*$ in Lemma 20.7.3.
$\square$
Lemma 20.13.8 (Relative Leray spectral sequence). Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. There is a spectral sequence with
\[ E_2^{p, q} = R^ pg_*(R^ qf_*\mathcal{F}) \]
converging to $R^{p + q}(g \circ f)_*\mathcal{F}$. This spectral sequence is functorial in $\mathcal{F}$, and there is a version for bounded below complexes of $\mathcal{O}_ X$-modules.
Proof.
This is a Grothendieck spectral sequence for composition of functors and follows from Lemma 20.13.7 and Derived Categories, Lemma 13.22.2.
$\square$
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