Example 21.42.1 (08RZ): Computing cohomology

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.

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