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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.


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