The Stacks project

Lemma 30.4.3. Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Then

  1. given a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ there exists an embedding $\mathcal{F} \to \mathcal{F}'$ of quasi-coherent $\mathcal{O}_ X$-modules such that $H^ p(X, \mathcal{F}') = 0$ for all $p \geq 1$, and

  2. $\{ H^ n(X, -)\} _{n \geq 0}$ is a universal $\delta $-functor from $\mathit{QCoh}(\mathcal{O}_ X)$ to $\textit{Ab}$.

Proof. Let $X = \bigcup U_ i$ be a finite affine open covering. Set $U = \coprod U_ i$ and denote $j : U \to X$ the morphism inducing the given open immersions $U_ i \to X$. Since $U$ is an affine scheme and $X$ has affine diagonal, the morphism $j$ is affine, see Morphisms, Lemma 29.11.11. For every $\mathcal{O}_ X$-module $\mathcal{F}$ there is a canonical map $\mathcal{F} \to j_*j^*\mathcal{F}$. This map is injective as can be seen by checking on stalks: if $x \in U_ i$, then we have a factorization

\[ \mathcal{F}_ x \to (j_*j^*\mathcal{F})_ x \to (j^*\mathcal{F})_{x'} = \mathcal{F}_ x \]

where $x' \in U$ is the point $x$ viewed as a point of $U_ i \subset U$. Now if $\mathcal{F}$ is quasi-coherent, then $j^*\mathcal{F}$ is quasi-coherent on the affine scheme $U$ hence has vanishing higher cohomology by Lemma 30.2.2. Then $H^ p(X, j_*j^*\mathcal{F}) = 0$ for $p > 0$ by Lemma 30.2.4 as $j$ is affine. This proves (1). Finally, we see that the map $H^ p(X, \mathcal{F}) \to H^ p(X, j_*j^*\mathcal{F})$ is zero and part (2) follows from Homology, Lemma 12.12.4. $\square$


Comments (2)

Comment #1843 by Keenan Kidwell on

In the statement, "diaganal" should be "diagonal."

There are also:

  • 3 comment(s) on Section 30.4: Quasi-coherence of higher direct images

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Your email address will not be published.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 0BDY. Beware of the difference between the letter 'O' and the digit '0'.