The Stacks project

Lemma 48.11.4. Let $f : Y \to X$ be a finite pseudo-coherent morphism of schemes (a finite morphism of Noetherian schemes is pseudo-coherent). The functor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)$ maps $D_\mathit{QCoh}^+(\mathcal{O}_ X)$ into $D_\mathit{QCoh}^+(f_*\mathcal{O}_ Y)$. If $X$ is quasi-compact and quasi-separated, then the diagram

\[ \xymatrix{ D_\mathit{QCoh}^+(\mathcal{O}_ X) \ar[rr]_ a \ar[rd]_{R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)} & & D_\mathit{QCoh}^+(\mathcal{O}_ Y) \ar[ld]^\Phi \\ & D_\mathit{QCoh}^+(f_*\mathcal{O}_ Y) } \]

is commutative, where $a$ is the right adjoint of Lemma 48.3.1 for $f$ and $\Phi $ is the equivalence of Derived Categories of Schemes, Lemma 36.5.4.

Proof. (The parenthetical remark follows from More on Morphisms, Lemma 37.60.9.) Since $f$ is pseudo-coherent, the $\mathcal{O}_ X$-module $f_*\mathcal{O}_ Y$ is pseudo-coherent, see More on Morphisms, Lemma 37.60.8. Thus $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)$ maps $D_\mathit{QCoh}^+(\mathcal{O}_ X)$ into $D_\mathit{QCoh}^+(f_*\mathcal{O}_ Y)$, see Derived Categories of Schemes, Lemma 36.10.8. Then $\Phi \circ a$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)$ agree on $D_\mathit{QCoh}^+(\mathcal{O}_ X)$ because these functors are both right adjoint to the restriction functor $D_\mathit{QCoh}^+(f_*\mathcal{O}_ Y) \to D_\mathit{QCoh}^+(\mathcal{O}_ X)$. To see this use Lemmas 48.3.5 and 48.11.2. $\square$

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