The Stacks project

Lemma 85.2.11. Let $X$ be a simplicial space and let $a : X \to Y$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X_{Zar}$. Then $R^ na_*\mathcal{F}$ is the sheaf associated to the presheaf

\[ V \longmapsto H^ n((X \times _ Y V)_{Zar}, \mathcal{F}|_{(X \times _ Y V)_{Zar}}) \]

Proof. This is the analogue of Cohomology, Lemma 20.7.3 or of Cohomology on Sites, Lemma 21.7.4 and we strongly encourge 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

\[ V \longmapsto H^0((X \times _ Y V)_{Zar}, \mathcal{F}|_{(X \times _ Y V)_{Zar}}) \]

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. $\square$

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