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. $\square$

Comment #1843 by Keenan Kidwell on

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

Comment #7194 by DatPham on

I guess the reason this is not a formal consequence of Gabber's result that the category $QCoh(\mathcal{O}_X)$ has enough injectives is just because the inclusion functor $QCoh(\mathcal{O}_X)\to Mod(\mathcal{O}_X)$ does not preserve injectives. Is this correct or am I missing something?

Comment #7320 by on

It is also not always when $X$ does not have affine diagonal.

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

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