The Stacks project

Remark 36.16.4. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ be a generator (see Theorem 36.15.3). Then the following are equivalent

1. for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $Rf_*K = 0$ if and only if $K = 0$,

2. $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ reflects isomorphisms, and

3. $Lf^*E$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$.

The equivalence between (1) and (2) is a formal consequence of the fact that $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that $Lf^*$ is the left adjoint to $Rf_*$. These conditions hold if $f$ is affine (Lemma 36.5.2) or if $f$ is an open immersion, or if $f$ is a composition of such. We conclude that

1. if $X$ is a quasi-affine scheme then $\mathcal{O}_ X$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$,

2. if $X \subset \mathbf{P}^ n_ A$ is a quasi-compact locally closed subscheme, then $\mathcal{O}_ X \oplus \mathcal{O}_ X(-1) \oplus \ldots \oplus \mathcal{O}_ X(-n)$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.16.3.