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21.14 The Leray spectral sequence

The key to proving the existence of the Leray spectral sequence is the following lemma.

Lemma 21.14.1. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Then for any injective object $\mathcal{I}$ in $\textit{Mod}(\mathcal{O}_\mathcal {C})$ the pushforward $f_*\mathcal{I}$ is totally acyclic.

Proof. Let $K$ be a sheaf of sets on $\mathcal{D}$. By Modules on Sites, Lemma 18.7.2 we may replace $\mathcal{C}$, $\mathcal{D}$ by “larger” sites such that $f$ comes from a morphism of ringed sites induced by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ such that $K = h_ V$ for some object $V$ of $\mathcal{D}$.

Thus we have to show that $H^ q(V, f_*\mathcal{I})$ is zero for $q > 0$ and all objects $V$ of $\mathcal{D}$ when $f$ is given by a morphism of ringed sites. Let $\mathcal{V} = \{ V_ j \to V\} $ be any covering of $\mathcal{D}$. Since $u$ is continuous we see that $\mathcal{U} = \{ u(V_ j) \to u(V)\} $ is a covering of $\mathcal{C}$. Then we have an equality of Čech complexes

\[ \check{\mathcal{C}}^\bullet (\mathcal{V}, f_*\mathcal{I}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}) \]

by the definition of $f_*$. By Lemma 21.12.3 we see that the cohomology of this complex is zero in positive degrees. We win by Lemma 21.10.9. $\square$

For flat morphisms the functor $f_*$ preserves injective modules. In particular the functor $f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$ always transforms injective abelian sheaves into injective abelian sheaves.

Lemma 21.14.2. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. If $f$ is flat, then $f_*\mathcal{I}$ is an injective $\mathcal{O}_\mathcal {D}$-module for any injective $\mathcal{O}_\mathcal {C}$-module $\mathcal{I}$.

Proof. In this case the functor $f^*$ is exact, see Modules on Sites, Lemma 18.31.2. Hence the result follows from Homology, Lemma 12.29.1. $\square$

Lemma 21.14.3. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be a ringed topos. A totally acyclic sheaf is right acyclic for the following functors:

  1. the functor $H^0(U, -)$ for any object $U$ of $\mathcal{C}$,

  2. the functor $\mathcal{F} \mapsto \mathcal{F}(K)$ for any presheaf of sets $K$,

  3. the functor $\Gamma (\mathcal{C}, -)$ of global sections,

  4. the functor $f_*$ for any morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ of ringed topoi.

Proof. Part (2) is the definition of a totally acyclic sheaf. Part (1) is a consequence of (2) as pointed out in the discussion following the definition of totally acyclic sheaves. Part (3) is a special case of (2) where $K = e$ is the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

To prove (4) we may assume, by Modules on Sites, Lemma 18.7.2 that $f$ is given by a morphism of sites. In this case we see that $R^ if_*$, $i > 0$ of a totally acyclic sheaf are zero by the description of higher direct images in Lemma 21.7.4. $\square$

Remark 21.14.4. As a consequence of the results above we find that Derived Categories, Lemma 13.22.1 applies to a number of situations. For example, given a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ of ringed topoi we have

\[ R\Gamma (\mathcal{D}, Rf_*\mathcal{F}) = R\Gamma (\mathcal{C}, \mathcal{F}) \]

for any sheaf of $\mathcal{O}_\mathcal {C}$-modules $\mathcal{F}$. Namely, for an injective $\mathcal{O}_\mathcal {X}$-module $\mathcal{I}$ the $\mathcal{O}_\mathcal {D}$-module $f_*\mathcal{I}$ is totally acyclic by Lemma 21.14.1 and a totally acyclic sheaf is acyclic for $\Gamma (\mathcal{D}, -)$ by Lemma 21.14.3.

Lemma 21.14.5 (Leray spectral sequence). Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_\mathcal {C}$-modules. There is a spectral sequence

\[ E_2^{p, q} = H^ p(\mathcal{D}, R^ qf_*(\mathcal{F}^\bullet )) \]

converging to $H^{p + q}(\mathcal{C}, \mathcal{F}^\bullet )$.

Proof. This is just the Grothendieck spectral sequence Derived Categories, Lemma 13.22.2 coming from the composition of functors $\Gamma (\mathcal{C}, -) = \Gamma (\mathcal{D}, -) \circ f_*$. To see that the assumptions of Derived Categories, Lemma 13.22.2 are satisfied, see Lemmas 21.14.1 and 21.14.3. $\square$

Lemma 21.14.6. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {C}$-module.

  1. If $R^ qf_*\mathcal{F} = 0$ for $q > 0$, then $H^ p(\mathcal{C}, \mathcal{F}) = H^ p(\mathcal{D}, f_*\mathcal{F})$ for all $p$.

  2. If $H^ p(\mathcal{D}, R^ qf_*\mathcal{F}) = 0$ for all $q$ and $p > 0$, then $H^ q(\mathcal{C}, \mathcal{F}) = H^0(\mathcal{D}, R^ qf_*\mathcal{F})$ for all $q$.

Proof. These are two simple conditions that force the Leray spectral sequence to converge. You can also prove these facts directly (without using the spectral sequence) which is a good exercise in cohomology of sheaves. $\square$

Lemma 21.14.7 (Relative Leray spectral sequence). Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ and $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{E}), \mathcal{O}_\mathcal {E})$ be morphisms of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {C}$-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 $\mathcal{O}_\mathcal {C}$-modules.

Proof. This is a Grothendieck spectral sequence for composition of functors, see Derived Categories, Lemma 13.22.2 and Lemmas 21.14.1 and 21.14.3. $\square$

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