Lemma 59.19.1. The forgetful functor $\textit{Ab}(\mathcal{C})\to \textit{PAb}(\mathcal{C})$ transforms injectives into injectives.
59.19 The Čech-to-cohomology spectral sequence
This spectral sequence is fundamental in proving foundational results on cohomology of sheaves.
Proof. This is formal using the fact that the forgetful functor has a left adjoint, namely sheafification, which is an exact functor. For more details see Cohomology on Sites, Lemma 21.10.1. $\square$
Theorem 59.19.2. Let $\mathcal{C}$ be a site. For any covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ there is a spectral sequence where $\underline{H}^ q(\mathcal{F})$ is the abelian presheaf $V \mapsto H^ q(V, \mathcal{F})$.
\[ E_2^{p, q} = \check H^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \Rightarrow H^{p+q}(U, \mathcal{F}), \]
Proof. Choose an injective resolution $\mathcal{F}\to \mathcal{I}^\bullet $ in $\textit{Ab}(\mathcal{C})$, and consider the double complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet )$ and the maps
\[ \xymatrix{ \Gamma (U, I^\bullet ) \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet ) \\ & \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \ar[u] } \]
Here the horizontal map is the natural map $\Gamma (U, I^\bullet ) \to \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^\bullet )$ to the left column, and the vertical map is induced by $\mathcal{F}\to \mathcal{I}^0$ and lands in the bottom row. By assumption, $\mathcal{I}^\bullet $ is a complex of injectives in $\textit{Ab}(\mathcal{C})$, hence by Lemma 59.19.1, it is a complex of injectives in $\textit{PAb}(\mathcal{C})$. Thus, the rows of the double complex are exact in positive degrees (Lemma 59.18.7), and the kernel of $\check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^\bullet ) \to \check{\mathcal{C}}^1(\mathcal{U}, \mathcal{I}^\bullet )$ is equal to $\Gamma (U, \mathcal{I}^\bullet )$, since $\mathcal{I}^\bullet $ is a complex of sheaves. In particular, the cohomology of the total complex is the standard cohomology of the global sections functor $H^0(U, \mathcal{F})$.
For the vertical direction, the $q$th cohomology group of the $p$th column is
\[ \prod _{i_0, \ldots , i_ p} H^ q(U_{i_0} \times _ U \ldots \times _ U U_{i_ p}, \mathcal{F}) = \prod _{i_0, \ldots , i_ p} \underline{H}^ q(\mathcal{F})(U_{i_0} \times _ U \ldots \times _ U U_{i_ p}) \]
in the entry $E_1^{p, q}$. So this is a standard double complex spectral sequence, and the $E_2$-page is as prescribed. For more details see Cohomology on Sites, Lemma 21.10.6. $\square$
Remark 59.19.3. This is a Grothendieck spectral sequence for the composition of functors
\[ \textit{Ab}(\mathcal{C}) \longrightarrow \textit{PAb}(\mathcal{C}) \xrightarrow {\check H^0} \textit{Ab}. \]
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