The Stacks project

Proof. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Z$-modules. Then

The first equality by Modules, Lemma 17.22.3 and the second by the fully faithfulness of $i_*$, see Modules, Lemma 17.13.4. The description of sections is left to the reader. $\square$

Proof. This is a consequence of the fact that $i_*$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ are adjoint functors by Lemma 48.9.1. See Derived Categories, Lemma 13.30.3. $\square$

Proof. This is immediate from the construction of the functor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$. $\square$

Proof. This is immediate from the construction of the functor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ and the fact that if $\mathcal{K}^\bullet$ is a K-injective complex of $\mathcal{O}_ X$-modules, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \mathcal{K}^\bullet )$ is a K-injective complex of $\mathcal{O}_ Z$-modules, see Derived Categories, Lemma 13.31.9. $\square$

Proof. To explain the parenthetical remark, if $X$ is locally Noetherian, then $i$ is pseudo-coherent by More on Morphisms, Lemma 37.60.9.

Let $K$ be an object of $D^+_\mathit{QCoh}(\mathcal{O}_ X)$. To prove (1), by Morphisms, Lemma 29.4.1 it suffices to show that $i_*$ applied to $H^ n(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, K))$ produces a quasi-coherent module on $X$. By Lemma 48.9.3 this means we have to show that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{O}_ Z, K)$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$. Since $i$ is pseudo-coherent the sheaf $\mathcal{O}_ Z$ is a pseudo-coherent $\mathcal{O}_ X$-module. Hence the result follows from Derived Categories of Schemes, Lemma 36.10.8.

Assume $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(B)$ as in (2). Let $I^\bullet$ be a bounded below complex of injective $A$-modules representing an object $K$ of $D^+(A)$. Then we know that $R\mathop{\mathrm{Hom}}\nolimits (B, K) = \mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet )$ viewed as a complex of $B$-modules. Choose a quasi-isomorphism

where $\mathcal{I}^\bullet$ is a bounded below complex of injective $\mathcal{O}_ X$-modules. It follows from the description of the functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ in Lemma 48.9.1 that there is a map

Observe that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \mathcal{I}^\bullet )$ represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \widetilde{K})$. Applying the universal property of the $\widetilde{\ }$ functor we obtain a map

in $D(\mathcal{O}_ Z)$. We may check that this map is an isomorphism in $D(\mathcal{O}_ Z)$ after applying $i_*$. However, once we apply $i_*$ we obtain the isomorphism of Derived Categories of Schemes, Lemma 36.10.8 via the identification of Lemma 48.9.3. $\square$

Proof. The question is local on $X$, hence we may assume that $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(B)$ with $A$ Noetherian and $A \to B$ surjective. In this case, we can apply Lemma 48.9.5 to translate the question into algebra. The corresponding algebra result is a consequence of Dualizing Complexes, Lemma 47.13.4. $\square$

Proof. (The parenthetical statement follows from More on Morphisms, Lemma 37.60.9.) 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. $\square$