21.19 Cohomology of unbounded complexes
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The category $\textit{Mod}(\mathcal{O})$ is a Grothendieck abelian category: it has all colimits, filtered colimits are exact, and it has a generator, namely
\[ \bigoplus \nolimits _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} j_{U!}\mathcal{O}_ U, \]
see Modules on Sites, Section 18.14 and Lemmas 18.28.7 and 18.28.8. By Injectives, Theorem 19.12.6 for every complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules there exists an injective quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ to a K-injective complex of $\mathcal{O}$-modules and moreover this embedding can be chosen functorial in $\mathcal{F}^\bullet $. It follows from Derived Categories, Lemma 13.31.7 that
any exact functor $F : K(\textit{Mod}(\mathcal{O})) \to \mathcal{D}$ into a trianguated category $\mathcal{D}$ has a right derived functor $RF : D(\mathcal{O}) \to \mathcal{D}$,
for any additive functor $F : \textit{Mod}(\mathcal{O}) \to \mathcal{A}$ into an abelian category $\mathcal{A}$ we consider the exact functor $F : K(\textit{Mod}(\mathcal{O})) \to D(\mathcal{A})$ induced by $F$ and we obtain a right derived functor $RF : D(\mathcal{O}) \to K(\mathcal{A})$.
By construction we have $RF(\mathcal{F}^\bullet ) = F(\mathcal{I}^\bullet )$ where $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ is as above.
Here are some examples of the above:
The functor $\Gamma (\mathcal{C}, -) : \textit{Mod}(\mathcal{O}) \to \text{Mod}_{\Gamma (\mathcal{C}, \mathcal{O})}$ gives rise to
\[ R\Gamma (\mathcal{C}, -) : D(\mathcal{O}) \longrightarrow D(\Gamma (\mathcal{C}, \mathcal{O})) \]
We shall use the notation $H^ i(\mathcal{C}, K) = H^ i(R\Gamma (\mathcal{C}, K))$ for cohomology.
For an object $U$ of $\mathcal{C}$ we consider the functor $\Gamma (U, -) : \textit{Mod}(\mathcal{O}) \to \text{Mod}_{\Gamma (U, \mathcal{O})}$. This gives rise to
\[ R\Gamma (U, -) : D(\mathcal{O}) \to D(\Gamma (U, \mathcal{O})) \]
We shall use the notation $H^ i(U, K) = H^ i(R\Gamma (U, K))$ for cohomology.
For a morphism of ringed topoi $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ we consider the functor $f_* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}')$ which gives rise to the total direct image
\[ Rf_* : D(\mathcal{O}) \longrightarrow D(\mathcal{O}') \]
on unbounded derived categories.
Lemma 21.19.1. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. The functor $Rf_*$ defined above and the functor $Lf^*$ defined in Lemma 21.18.2 are adjoint:
\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(Lf^*\mathcal{G}^\bullet , \mathcal{F}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}')}(\mathcal{G}^\bullet , Rf_*\mathcal{F}^\bullet ) \]
bifunctorially in $\mathcal{F}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}))$ and $\mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}'))$.
Proof.
This follows formally from the fact that $Rf_*$ and $Lf^*$ exist, see Derived Categories, Lemma 13.30.3.
$\square$
Lemma 21.19.2. 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. Then $Rg_* \circ Rf_* = R(g \circ f)_*$ as functors $D(\mathcal{O}_\mathcal {C}) \to D(\mathcal{O}_\mathcal {E})$.
Proof.
By Lemma 21.19.1 we see that $Rg_* \circ Rf_*$ is adjoint to $Lf^* \circ Lg^*$. We have $Lf^* \circ Lg^* = L(g \circ f)^*$ by Lemma 21.18.3 and hence by uniqueness of adjoint functors we have $Rg_* \circ Rf_* = R(g \circ f)_*$.
$\square$
Lemma 21.19.6. 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{K}^\bullet $ be a complex of $\mathcal{O}_\mathcal {C}$-modules. The diagram
\[ \xymatrix{ Lf^*f_*\mathcal{K}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \\ Lf^*Rf_*\mathcal{K}^\bullet \ar[r] & \mathcal{K}^\bullet } \]
coming from $Lf^* \to f^*$ on complexes, $f_* \to Rf_*$ on complexes, and adjunction $Lf^* \circ Rf_* \to \text{id}$ commutes in $D(\mathcal{O}_\mathcal {C})$.
Proof.
We will use the existence of K-flat resolutions and K-injective resolutions, see Lemmas 21.17.11, 21.18.2, and 21.18.1 and the discussion above. Choose a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{I}^\bullet $ where $\mathcal{I}^\bullet $ is K-injective as a complex of $\mathcal{O}_\mathcal {C}$-modules. Choose a quasi-isomorphism $\mathcal{Q}^\bullet \to f_*\mathcal{I}^\bullet $ where $\mathcal{Q}^\bullet $ is a K-flat complex of $\mathcal{O}_\mathcal {D}$-modules with flat terms. We can choose a K-flat complex of $\mathcal{O}_\mathcal {D}$-modules $\mathcal{P}^\bullet $ with flat terms and a diagram of morphisms of complexes
\[ \xymatrix{ \mathcal{P}^\bullet \ar[r] \ar[d] & f_*\mathcal{K}^\bullet \ar[d] \\ \mathcal{Q}^\bullet \ar[r] & f_*\mathcal{I}^\bullet } \]
commutative up to homotopy where the top horizontal arrow is a quasi-isomorphism. Namely, we can first choose such a diagram for some complex $\mathcal{P}^\bullet $ because the quasi-isomorphisms form a multiplicative system in the homotopy category of complexes and then we can choose a resolution of $\mathcal{P}^\bullet $ by a K-flat complex with flat terms. Taking pullbacks we obtain a diagram of morphisms of complexes
\[ \xymatrix{ f^*\mathcal{P}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \ar[r] & \mathcal{K}^\bullet \ar[d] \\ f^*\mathcal{Q}^\bullet \ar[r] & f^*f_*\mathcal{I}^\bullet \ar[r] & \mathcal{I}^\bullet } \]
commutative up to homotopy. The outer rectangle witnesses the truth of the statement in the lemma.
$\square$
Lemma 21.19.8. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \subset \textit{Ab}(\mathcal{C})$ denote the Serre subcategory consisting of torsion abelian sheaves. Then the functor $D(\mathcal{A}) \to D_\mathcal {A}(\mathcal{C})$ is an equivalence.
Proof.
A key observation is that an injective abelian sheaf $\mathcal{I}$ is divisible. Namely, if $s \in \mathcal{I}(U)$ is a local section, then we interpret $s$ as a map $s : j_{U!}\mathbf{Z} \to \mathcal{I}$ and we apply the defining property of an injective object to the injective map of sheaves $n : j_{U!}\mathbf{Z} \to j_{U!}\mathbf{Z}$ to see that there exists an $s' \in \mathcal{I}(U)$ with $ns' = s$.
For a sheaf $\mathcal{F}$ denote $\mathcal{F}_{tor}$ its torsion subsheaf. We claim that if $\mathcal{I}^\bullet $ is a complex of injective abelian sheaves whose cohomology sheaves are torsion, then
\[ \mathcal{I}^\bullet _{tor} \to \mathcal{I}^\bullet \]
is a quasi-isomorphism. Namely, by flatness of $\mathbf{Q}$ over $\mathbf{Z}$ we have
\[ H^ p(\mathcal{I}^\bullet ) \otimes _\mathbf {Z} \mathbf{Q} = H^ p(\mathcal{I}^\bullet \otimes _\mathbf {Z} \mathbf{Q}) \]
which is zero because the cohomology sheaves are torsion. By divisibility (shown above) we see that $\mathcal{I}^\bullet \to \mathcal{I}^\bullet \otimes _\mathbf {Z} \mathbf{Q}$ is surjective with kernel $\mathcal{I}^\bullet _{tor}$. The claim follows from the long exact sequence of cohomology sheaves associated to the short exact sequence you get.
To prove the lemma we will construct right adjoint $T : D(\mathcal{C}) \to D(\mathcal{A})$. Namely, given $K$ in $D(\mathcal{C})$ we can represent $K$ by a K-injective complex $\mathcal{I}^\bullet $ whose cohomology sheaves are injective, see Injectives, Theorem 19.12.6. Then we set $T(K) = \mathcal{I}^\bullet _{tor}$, in other words, $T$ is the right derived functor of taking torsion. The functor $T$ is a right adjoint to $i : D(\mathcal{A}) \to D_\mathcal {A}(\mathcal{C})$. This readily follows from the observation that if $\mathcal{F}^\bullet $ is a complex of torsion sheaves, then
\[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(\mathcal{F}^\bullet , I^\bullet _{tor}) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Ab}(\mathcal{C}))}(\mathcal{F}^\bullet , I^\bullet ) \]
in particular $\mathcal{I}^\bullet _{tor}$ is a K-injective complex of $\mathcal{A}$. Some details omitted; in case of doubt, it also follows from the more general Derived Categories, Lemma 13.30.3. Our claim above gives that $L = T(i(L))$ for $L$ in $D(\mathcal{A})$ and $i(T(K)) = K$ if $K$ is in $D_\mathcal {A}(\mathcal{C})$. Using Categories, Lemma 4.24.4 the result follows.
$\square$
Comments (0)