Lemma 91.16.3 (0DIV)—The Stacks project

Lemma 91.16.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\alpha : K \to L$ be a map of $D^-(\mathcal{O})$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $n \in \mathbf{Z}$.

If $H^ i(\alpha )$ is an isomorphism for $i \geq n$, then $H^ i(\alpha \otimes _\mathcal {O}^\mathbf {L} \text{id}_\mathcal {F})$ is an isomorphism for $i \geq n$.
If $H^ i(\alpha )$ is an isomorphism for $i > n$ and surjective for $i = n$, then $H^ i(\alpha \otimes _\mathcal {O}^\mathbf {L} \text{id}_\mathcal {F})$ is an isomorphism for $i > n$ and surjective for $i = n$.

Proof. Choose a distinguished triangle

\[ K \to L \to C \to K[1] \]

In case (2) we see that $H^ i(C) = 0$ for $i \geq n$. Hence $H^ i(C \otimes _\mathcal {O}^\mathbf {L} \mathcal{F}) = 0$ for $i \geq n$ by (the dual of) Derived Categories, Lemma 13.16.1. This in turn shows that $H^ i(\alpha \otimes _\mathcal {O}^\mathbf {L} \text{id}_\mathcal {F})$ is an isomorphism for $i > n$ and surjective for $i = n$. In case (1) we moreover see that $H^{n - 1}(L) \to H^{n - 1}(C)$ is surjective. Considering the diagram

\[ \xymatrix{ H^{n - 1}(L) \otimes _\mathcal {O} \mathcal{F} \ar[r] \ar[d] & H^{n - 1}(C) \otimes _\mathcal {O} \mathcal{F} \ar@{=}[d] \\ H^{n - 1}(L \otimes _\mathcal {O}^\mathbf {L} \mathcal{F}) \ar[r] & H^{n - 1}(C \otimes _\mathcal {O}^\mathbf {L} \mathcal{F}) } \]

we conclude the lower horizontal arrow is surjective. Combined with what was said before this implies that $H^ n(\alpha \otimes _\mathcal {O}^\mathbf {L} \text{id}_\mathcal {F})$ is an isomorphism. $\square$

The Stacks project

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