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48.13 Right adjoint of pushforward for perfect proper morphisms

The correct generality for this section would be to consider perfect proper morphisms of quasi-compact and quasi-separated schemes, see [LN].

Lemma 48.13.1. Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. Let $a$ be the right adjoint for $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ of Lemma 48.3.1. Then $a$ commutes with direct sums.

Proof. Let $P$ be a perfect object of $D(\mathcal{O}_ X)$. By More on Morphisms, Lemma 37.53.12 the complex $Rf_*P$ is perfect on $Y$. Let $K_ i$ be a family of objects of $D_\mathit{QCoh}(\mathcal{O}_ Y)$. Then

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P, a(\bigoplus K_ i)) & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}(Rf_*P, \bigoplus K_ i) \\ & = \bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}(Rf_*P, K_ i) \\ & = \bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P, a(K_ i)) \end{align*}

because a perfect object is compact (Derived Categories of Schemes, Proposition 36.16.1). Since $D_\mathit{QCoh}(\mathcal{O}_ X)$ has a perfect generator (Derived Categories of Schemes, Theorem 36.14.3) we conclude that the map $\bigoplus a(K_ i) \to a(\bigoplus K_ i)$ is an isomorphism, i.e., $a$ commutes with direct sums. $\square$

Lemma 48.13.2. Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. Let $a$ be the right adjoint for $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ of Lemma 48.3.1. Then

  1. for every closed $T \subset Y$ if $Q \in D_\mathit{QCoh}(Y)$ is supported on $T$, then $a(Q)$ is supported on $f^{-1}(T)$,

  2. for every open $V \subset Y$ and any $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ the map (48.4.1.1) is an isomorphism, and

Lemma 48.13.3. Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. The map (48.8.0.1) is an isomorphism for every object $K$ of $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. By Lemma 48.13.1 we know that $a$ commutes with direct sums. Hence the collection of objects of $D_\mathit{QCoh}(\mathcal{O}_ Y)$ for which (48.8.0.1) is an isomorphism is a strictly full, saturated, triangulated subcategory of $D_\mathit{QCoh}(\mathcal{O}_ Y)$ which is moreover preserved under taking direct sums. Since $D_\mathit{QCoh}(\mathcal{O}_ Y)$ is a module category (Derived Categories of Schemes, Theorem 36.17.3) generated by a single perfect object (Derived Categories of Schemes, Theorem 36.14.3) we can argue as in More on Algebra, Remark 15.57.13 to see that it suffices to prove (48.8.0.1) is an isomorphism for a single perfect object. However, the result holds for perfect objects, see Lemma 48.8.1. $\square$

Lemma 48.13.4. Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. Let $g : Y' \to Y$ be a morphism with $Y'$ Noetherian. If $X$ and $Y'$ are tor independent over $Y$, then the base change map (48.5.0.1) is an isomorphism for all $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. By Lemma 48.13.2 formation of the functors $a$ and $a'$ commutes with restriction to opens of $Y$ and $Y'$. Hence we may assume $Y' \to Y$ is a morphism of affine schemes, see Remark 48.6.1. In this case the statement follows from Lemma 48.6.2. $\square$


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