The Stacks project

Proof. This is the analogue of Cohomology, Lemma 20.7.3 or of Cohomology on Sites, Lemma 21.7.4 and we strongly encourage the reader to skip the proof. Choosing an injective resolution of $\mathcal{F}$ on $X_{Zar}$ and using the definitions we see that it suffices to show: (1) the restriction of an injective abelian sheaf on $X_{Zar}$ to $(X \times _ Y V)_{Zar}$ is an injective abelian sheaf and (2) $a_*\mathcal{F}$ is equal to the rule

Part (2) follows from the following facts

1. $a_*\mathcal{F}$ is the equalizer of the two maps $a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1$ by Lemma 85.2.8,

2. $a_{0, *}\mathcal{F}_0(V) = H^0(a_0^{-1}(V), \mathcal{F}_0)$ and $a_{1, *}\mathcal{F}_1(V) = H^0(a_1^{-1}(V), \mathcal{F}_1)$,

3. $X_0 \times _ Y V = a_0^{-1}(V)$ and $X_1 \times _ Y V = a_1^{-1}(V)$,

4. $H^0((X \times _ Y V)_{Zar}, \mathcal{F}|_{(X \times _ Y V)_{Zar}})$ is the equalizer of the two maps $H^0(X_0 \times _ Y V, \mathcal{F}_0) \to H^0(X_1 \times _ Y V, \mathcal{F}_1)$ for example by Lemma 85.2.10.

Part (1) follows after one defines an exact left adjoint $j_! : \textit{Ab}((X \times _ Y V)_{Zar}) \to \textit{Ab}(X_{Zar})$ (extension by zero) to restriction $\textit{Ab}(X_{Zar}) \to \textit{Ab}((X \times _ Y V)_{Zar})$ and using Homology, Lemma 12.29.1. $\square$