The Stacks project

Lemma 48.4.3. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a$ be the right adjoint to $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. Let $V \subset Y$ be quasi-compact open with inverse image $U \subset X$.

  1. For every $Q \in D_\mathit{QCoh}^+(\mathcal{O}_ Y)$ supported on $Y \setminus V$ the image $a(Q)$ is supported on $X \setminus U$ if and only if ( is an isomorphism on all $K$ in $D_\mathit{QCoh}^+(\mathcal{O}_ Y)$.

  2. For every $Q \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ supported on $Y \setminus V$ the image $a(Q)$ is supported on $X \setminus U$ if and only if ( is an isomorphism on all $K$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

  3. If $a$ commutes with direct sums, then the equivalent conditions of (1) imply the equivalent conditions of (2).

Proof. Proof of (1). Let $K \in D_\mathit{QCoh}^+(\mathcal{O}_ Y)$. Choose a distinguished triangle

\[ K \to Rj_*K|_ V \to Q \to K[1] \]

Observe that $Q$ is in $D_\mathit{QCoh}^+(\mathcal{O}_ Y)$ (Derived Categories of Schemes, Lemma 36.4.1) and is supported on $Y \setminus V$ (Derived Categories of Schemes, Definition 36.6.1). Applying $a$ we obtain a distinguished triangle

\[ a(K) \to a(Rj_*K|_ V) \to a(Q) \to a(K)[1] \]

on $X$. If $a(Q)$ is supported on $X \setminus U$, then restricting to $U$ the map $a(K)|_ U \to a(Rj_*K|_ V)|_ U$ is an isomorphism, i.e., ( is an isomorphism on $K$. The converse is immediate.

The proof of (2) is exactly the same as the proof of (1).

Proof of (3). Assume the equivalent conditions of (1) hold. Set $T = Y \setminus V$. We will use the notation $D_{\mathit{QCoh}, T}(\mathcal{O}_ Y)$ and $D_{\mathit{QCoh}, f^{-1}(T)}(\mathcal{O}_ X)$ to denote complexes whose cohomology sheaves are supported on $T$ and $f^{-1}(T)$. Since $a$ commutes with direct sums, the strictly full, saturated, triangulated subcategory $\mathcal{D}$ with objects

\[ \{ Q \in D_{\mathit{QCoh}, T}(\mathcal{O}_ Y) \mid a(Q) \in D_{\mathit{QCoh}, f^{-1}(T)}(\mathcal{O}_ X)\} \]

is preserved by direct sums and hence derived colimits. On the other hand, the category $D_{\mathit{QCoh}, T}(\mathcal{O}_ Y)$ is generated by a perfect object $E$ (see Derived Categories of Schemes, Lemma 36.15.4). By assumption we see that $E \in \mathcal{D}$. By Derived Categories, Lemma 13.37.3 every object $Q$ of $D_{\mathit{QCoh}, T}(\mathcal{O}_ Y)$ is a derived colimit of a system $Q_1 \to Q_2 \to Q_3 \to \ldots $ such that the cones of the transition maps are direct sums of shifts of $E$. Arguing by induction we see that $Q_ n \in \mathcal{D}$ for all $n$ and finally that $Q$ is in $\mathcal{D}$. Thus the equivalent conditions of (2) hold. $\square$

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