Lemma 48.11.4 (0AX2)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.11: Right adjoint of pushforward for finite morphisms
Lemma 48.11.4 (cite)

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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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