21.42 Cohomology on a category
In the situation of Example 21.39.1 in addition to the derived functor $L\pi _!$, we also have the functor $R\pi _*$. For an abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ we have $H_ n(\mathcal{C}, \mathcal{F}) = H^{-n}(L\pi _!\mathcal{F})$ and $H^ n(\mathcal{C}, \mathcal{F}) = H^ n(R\pi _*\mathcal{F})$.
Example 21.42.1 (Computing cohomology). In Example 21.39.1 we can compute the functors $H^ n(\mathcal{C}, -)$ as follows. Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Ab}(\mathcal{C}))$. Consider the cochain complex
\[ K^\bullet (\mathcal{F}) : \prod \nolimits _{U_0} \mathcal{F}(U_0) \to \prod \nolimits _{U_0 \to U_1} \mathcal{F}(U_0) \to \prod \nolimits _{U_0 \to U_1 \to U_2} \mathcal{F}(U_0) \to \ldots \]
where the transition maps are given by
\[ (s_{U_0 \to U_1}) \longmapsto ((U_0 \to U_1 \to U_2) \mapsto s_{U_0 \to U_1} - s_{U_0 \to U_2} + s_{U_1 \to U_2}|_{U_0}) \]
and similarly in other degrees. By construction
\[ H^0(\mathcal{C}, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits _{\mathcal{C}^{opp}} \mathcal{F} = H^0(K^\bullet (\mathcal{F})), \]
see Categories, Lemma 4.14.11. The construction of $K^\bullet (\mathcal{F})$ is functorial in $\mathcal{F}$ and transforms short exact sequences of $\textit{Ab}(\mathcal{C})$ into short exact sequences of complexes. Thus the sequence of functors $\mathcal{F} \mapsto H^ n(K^\bullet (\mathcal{F}))$ forms a $\delta $-functor, see Homology, Definition 12.12.1 and Lemma 12.13.12. For an object $U$ of $\mathcal{C}$ denote $p_ U : \mathop{\mathit{Sh}}\nolimits (*) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ the corresponding point with $p_ U^{-1}$ equal to evaluation at $U$, see Sites, Example 7.33.8. Let $A$ be an abelian group and set $\mathcal{F} = p_{U, *}A$. In this case the complex $K^\bullet (\mathcal{F})$ is the complex with terms $\text{Map}(X_ n, A)$ where
\[ X_ n = \coprod \nolimits _{U_0 \to \ldots \to U_{n - 1} \to U_ n} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, U_0) \]
This simplicial set is homotopy equivalent to the constant simplicial set on a singleton $\{ *\} $. Namely, the map $X_\bullet \to \{ *\} $ is obvious, the map $\{ *\} \to X_ n$ is given by mapping $*$ to $(U \to \ldots \to U, \text{id}_ U)$, and the maps
\[ h_{n, i} : X_ n \longrightarrow X_ n \]
(Simplicial, Lemma 14.26.2) defining the homotopy between the two maps $X_\bullet \to X_\bullet $ are given by the rule
\[ h_{n, i} : (U_0 \to \ldots \to U_ n, f) \longmapsto (U \to \ldots \to U \to U_ i \to \ldots \to U_ n, \text{id}) \]
for $i > 0$ and $h_{n, 0} = \text{id}$. Verifications omitted. Since $\text{Map}(-, A)$ is a contravariant functor, implies that $K^\bullet (p_{U, *}A)$ has trivial cohomology in positive degrees (by the functoriality of Simplicial, Remark 14.26.4 and the result of Simplicial, Lemma 14.28.6). This implies that $K^\bullet (\mathcal{F})$ is acyclic in positive degrees also if $\mathcal{F}$ is a product of sheaves of the form $p_{U, *}A$. As every abelian sheaf on $\mathcal{C}$ embeds into such a product we conclude that $K^\bullet (\mathcal{F})$ computes the left derived functors $H^ n(\mathcal{C}, -)$ of $H^0(\mathcal{C}, -)$ for example by Homology, Lemma 12.12.4 and Derived Categories, Lemma 13.16.6.
Example 21.42.2 (Computing Exts). In Example 21.39.1 assume we are moreover given a sheaf of rings $\mathcal{O}$ on $\mathcal{C}$. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}$-modules. Consider the complex $K^\bullet (\mathcal{G}, \mathcal{F})$ with degree $n$ term
\[ \prod \nolimits _{U_0 \to U_1 \to \ldots \to U_ n} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}(U_ n)}(\mathcal{G}(U_ n), \mathcal{F}(U_0)) \]
and transition map given by
\[ (\varphi _{U_0 \to U_1}) \longmapsto ((U_0 \to U_1 \to U_2) \mapsto \varphi _{U_0 \to U_1} \circ \rho ^{U_2}_{U_1} - \varphi _{U_0 \to U_2} + \rho ^{U_1}_{U_0} \circ \varphi _{U_1 \to U_2} \]
and similarly in other degrees. Here the $\rho $'s indicate restriction maps. By construction
\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{F}) = H^0(K^\bullet (\mathcal{G}, \mathcal{F})) \]
for all pairs of $\mathcal{O}$-modules $\mathcal{F}, \mathcal{G}$. The assignment $(\mathcal{G}, \mathcal{F}) \mapsto K^\bullet (\mathcal{G}, \mathcal{F})$ is a bifunctor which transforms direct sums in the first variable into products and commutes with products in the second variable. We claim that
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {O}(\mathcal{G}, \mathcal{F}) = H^ i(K^\bullet (\mathcal{G}, \mathcal{F})) \]
for $i \geq 0$ provided either
$\mathcal{G}(U)$ is a projective $\mathcal{O}(U)$-module for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, or
$\mathcal{F}(U)$ is an injective $\mathcal{O}(U)$-module for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.
Namely, case (1) the functor $K^\bullet (\mathcal{G}, -)$ is an exact functor from the category of $\mathcal{O}$-modules to the category of cochain complexes of abelian groups. Thus, arguing as in Example 21.42.1, it suffices to show that $K^\bullet (\mathcal{G}, \mathcal{F})$ is acyclic in positive degrees when $\mathcal{F}$ is $p_{U, *}A$ for an $\mathcal{O}(U)$-module $A$. Choose a short exact sequence
21.42.2.1
\begin{equation} \label{sites-cohomology-equation-split} 0 \to \mathcal{G}' \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \to \mathcal{G} \to 0 \end{equation}
see Modules on Sites, Lemma 18.28.8. Since (1) holds for the middle and right sheaves, it also holds for $\mathcal{G}'$ and evaluating (21.42.2.1) on an object of $\mathcal{C}$ gives a split exact sequence of modules. We obtain a short exact sequence of complexes
\[ 0 \to K^\bullet (\mathcal{G}, \mathcal{F}) \to \prod K^\bullet (j_{U_ i!}\mathcal{O}_{U_ i}, \mathcal{F}) \to K^\bullet (\mathcal{G}', \mathcal{F}) \to 0 \]
for any $\mathcal{F}$, in particular $\mathcal{F} = p_{U, *}A$. On $H^0$ we obtain
\[ 0 \to \mathop{\mathrm{Hom}}\nolimits (\mathcal{G}, p_{U, *}A) \to \mathop{\mathrm{Hom}}\nolimits (\prod j_{U_ i!}\mathcal{O}_{U_ i}, p_{U, *}A) \to \mathop{\mathrm{Hom}}\nolimits (\mathcal{G}', p_{U, *}A) \to 0 \]
which is exact as $\mathop{\mathrm{Hom}}\nolimits (\mathcal{H}, p_{U, *}A) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}(U)}(\mathcal{H}(U), A)$ and the sequence of sections of (21.42.2.1) over $U$ is split exact. Thus we can use dimension shifting to see that it suffices to prove $K^\bullet (j_{U'!}\mathcal{O}_{U'}, p_{U, *}A)$ is acyclic in positive degrees for all $U, U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. In this case $K^ n(j_{U'!}\mathcal{O}_{U'}, p_{U, *}A)$ is equal to
\[ \prod \nolimits _{U \to U_0 \to U_1 \to \ldots \to U_ n \to U'} A \]
In other words, $K^\bullet (j_{U'!}\mathcal{O}_{U'}, p_{U, *}A)$ is the complex with terms $\text{Map}(X_\bullet , A)$ where
\[ X_ n = \coprod \nolimits _{U_0 \to \ldots \to U_{n - 1} \to U_ n} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, U_0) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_ n, U') \]
This simplicial set is homotopy equivalent to the constant simplicial set on a singleton $\{ *\} $ as can be proved in exactly the same way as the corresponding statement in Example 21.42.1. This finishes the proof of the claim.
The argument in case (2) is similar (but dual).
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