The Stacks project

Lemma 48.10.3. Let $i : Z \to X$ be a pseudo-coherent closed immersion of schemes. Let $M \in D_\mathit{QCoh}(\mathcal{O}_ X)$ locally have tor-amplitude in $[a, \infty )$. Let $K \in D_\mathit{QCoh}^+(\mathcal{O}_ X)$. Then there is a canonical isomorphism

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, K) \otimes _{\mathcal{O}_ Z}^\mathbf {L} Li^*M = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \]

in $D(\mathcal{O}_ Z)$.

Proof. A map from LHS to RHS is the same thing as a map

\[ Ri_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, K) \otimes _{\mathcal{O}_ X}^\mathbf {L} M \longrightarrow K \otimes _{\mathcal{O}_ X}^\mathbf {L} M \]

by Lemmas 48.9.2 and 48.9.3. For this map we take the counit $Ri_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, K) \to K$ tensored with $\text{id}_ M$. To see this map is an isomorphism under the hypotheses given, translate into algebra using Lemma 48.9.5 and then for example use More on Algebra, Lemma 15.98.3 part (3). Instead of using Lemma 48.9.5 you can look at stalks as in the second proof of Lemma 48.10.2. $\square$

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