The Stacks project

Remark 75.15.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ be a generator (see Theorem 75.15.4). 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 75.6.4) or if $f$ is an open immersion, or if $f$ is a composition of such.