The Stacks project

Lemma 48.9.5. Let $i : Z \to X$ be a pseudo-coherent closed immersion of schemes (any closed immersion if $X$ is locally Noetherian). Then

  1. $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ maps $D^+_\mathit{QCoh}(\mathcal{O}_ X)$ into $D^+_\mathit{QCoh}(\mathcal{O}_ Z)$, and

  2. if $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(B)$, then the diagram

    \[ \xymatrix{ D^+(B) \ar[r] & D_\mathit{QCoh}^+(\mathcal{O}_ Z) \\ D^+(A) \ar[r] \ar[u]^{R\mathop{\mathrm{Hom}}\nolimits (B, -)} & D_\mathit{QCoh}^+(\mathcal{O}_ X) \ar[u]_{R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)} } \]

    is commutative.

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

\[ \widetilde{I^\bullet } \longrightarrow \mathcal{I}^\bullet \]

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

\[ \mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet ) \longrightarrow \Gamma (Z, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \mathcal{I}^\bullet )) \]

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

\[ \widetilde{\mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet )} \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \widetilde{K}) \]

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. $\square$

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