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Let $QCoh(X)$ be the category of Quasi coherent sheaves on a scheme $ X$ . If $X$ is affine and $\mathcal F \in QCoh(X)$ then it is known that the higher sheaf cohomologies are all zero i.e. $H^j(X,\mathcal F)=0,\forall j >0$. (One doesn't need $X$ to be Noetherian, see Theorem 2 of this paper by Kempf Link ) .

Now I have a question, if true, would possibly provide another proof of the above vanishing result of Serre: Let $X$ be a scheme. Suppose the global section functor $\Gamma : QCoh(X) \to Ab$ is exact. Then is $H^j (X, \mathcal F)=0,\forall j >0$ ?

The way I think a positive answer to this would provide a proof of the vanishing result is as follows: Let $X=$Spec $(R)$ be affine. If $\mathcal F$ is a Quasi coherent sheaf then $\mathcal F \cong \tilde M$ for some $R$-module $M$ . I claim that the global section functor on $QCoh(X)$ is exact. Indeed let $0\to \tilde M_1 \to \tilde M_2 \to \tilde M_3 \to 0$ be exact in $QCoh(X)$ , then since $\tilde M_x=M_P$ where $x$ is the point $P$ of Spec $(R)$, localising we get $0\to (M_1)_P \to (M_2)_P \to (M_3)_P \to 0$ is exact for every $P \in$ Spec $(R)$, but then $0\to M_1 \to M_2 \to M_3 \to 0$ is exact and since $\Gamma(\tilde M)=M$ this proves the exactness of the Global section functor on $QCoh(X)$ .

I hope I'm not missing anything ...

Glorfindel
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user
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    Sheaf cohomology is not the derived functor of $QCoh(X)\to Ab$, but the derived functor $\mathcal{O}_X-Mod\to Ab$. Even if the inclusion $QCoh(X)\to\mathcal{O}_X-Mod$ is exact, the derived functor may be different. See this answer : https://math.stackexchange.com/questions/2550534/godement-resolution-dont-see-mathcalo-x-module-structure/2563845#2563845. The key point of the proof is to show that an injective quasi-coherent sheaf on $\operatorname{Spec}R$ is flabby (though not injective as an $\mathcal{O}_X$-module). Nevertheless, they can be used to compute the derived functor of $\Gamma$. – Roland Sep 18 '19 at 08:59
  • @Roland: thanks for the link, it was very helpful ... however the slink still doesn't give an explicit example for the scheme case ... I would really like an explicit example where my question has negative answer ... – user Sep 18 '19 at 17:16

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