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.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BQT. Beware of the difference between the letter 'O' and the digit '0'.