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.
21.14 The Leray spectral sequence
The key to proving the existence of the Leray spectral sequence is the following lemma.
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:
the functor $H^0(U, -)$ for any object $U$ of $\mathcal{C}$,
the functor $\mathcal{F} \mapsto \mathcal{F}(K)$ for any presheaf of sets $K$,
the functor $\Gamma (\mathcal{C}, -)$ of global sections,
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 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.
\[ R\Gamma (\mathcal{D}, Rf_*\mathcal{F}) = R\Gamma (\mathcal{C}, \mathcal{F}) \]
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 converging to $H^{p + q}(\mathcal{C}, \mathcal{F}^\bullet )$.
\[ E_2^{p, q} = H^ p(\mathcal{D}, R^ qf_*(\mathcal{F}^\bullet )) \]
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.
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 $p$.
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 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.
\[ E_2^{p, q} = R^ pg_*(R^ qf_*\mathcal{F}) \]
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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)