The Stacks project

Lemma 48.9.7. Let $X$ be a quasi-compact and quasi-separated scheme. Let $i : Z \to X$ be a pseudo-coherent closed immersion (if $X$ is Noetherian, then any closed immersion is pseudo-coherent). Let $a : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Z)$ be the right adjoint to $Ri_*$. Then there is a functorial isomorphism

\[ a(K) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, K) \]

for $K \in D_\mathit{QCoh}^+(\mathcal{O}_ X)$.

Proof. (The parenthetical statement follows from More on Morphisms, Lemma 37.52.9.) By Lemma 48.9.2 the functor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ is a right adjoint to $Ri_* : D(\mathcal{O}_ Z) \to D(\mathcal{O}_ X)$. Moreover, by Lemma 48.9.5 and Lemma 48.3.5 both $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ and $a$ map $D_\mathit{QCoh}^+(\mathcal{O}_ X)$ into $D_\mathit{QCoh}^+(\mathcal{O}_ Z)$. Hence we obtain the isomorphism by uniqueness of adjoint functors. $\square$


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