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In Qing Liu's "Algebraic Geometry and Arithmetic Curves", page 161, Definition 1.10. Let $(X,\mathcal O_X)$ be a ringed topological space, and let $\mathcal F$ be an $\mathcal O_X$-module. We say that $\mathcal F$ is finitely generated If for every $x\in X$, there exist an open neighborhood $U$ of $x$, an integer $n\ge 1$, and a surjective homomorphism $\mathcal O^n_X|_U\to \mathcal F|_U$.

In page 162, Proposition 1.12. Let $X$ be a scheme. We have the following properties for $\mathcal O_X$-modules. (a) A direct sum of quasi-coherent sheaves is quasi-coherent; a finite direct sum of finitely generated quasi-coherent sheaves is finitely generated.

In the last half of Proposition 1.12 (a)--"a finite direct sum of finitely generated quasi-coherent sheaves is finitely generated", is the condition "quasi-coherent" necessary?

Now I'll show that a finite direct sum of finitely generated sheaves is finitely generated.

Let $\mathcal F_i(i=1,2,\cdots,k)$ be finitely generated $\mathcal O_X$-modules.

For every $x\in X$, there exist an open neighborhood $U_i$ of $x$, an integer $n_i\ge 1$, and a surjective homomorphism $\alpha_i:\mathcal O^{n_i}_X|_{U_i}\to \mathcal F_i|_{U_i}$.

Let $U=\bigcap_{i=1,2,\cdots,k}U_i$. We define $\oplus_{i=1,2,\cdots,k}\alpha_i:\mathcal O^{n_1+n_2+\cdots+n_k}_X|_U\to (\bigoplus_{i=1,2,\cdots,k}\mathcal F_i)|_U$ by $(\oplus_{i=1,2,\cdots,k}\alpha_i)(V):(a_1,a_2,\cdots,a_k)\mapsto (\alpha_1(V)(a_1),\alpha_2(V)(a_2),\cdots,\alpha_k(V)(a_k)),a_i\in \mathcal O^{n_i}_X(V),\textrm{open set $V\subset U$}.$

$\forall x\in U, ((\bigoplus_{i=1,2,\cdots,k}\mathcal{F}_i)|_U)_x\simeq \bigoplus_{i=1,2,\cdots,k}(\mathcal{F}_i|_U)_x.$

Then $\oplus_{i=1,2,\cdots,k}\alpha_i$ is surjective, so a finite direct sum of finitely generated $\mathcal O_X$-modules is finitely generated.

Is my argument correct?

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  • $\begingroup$ @LiamKeenan How to show "In particular, this means that $\mathcal F$ is quasi-coherent"? $\endgroup$ – Born to be proud Dec 10 '17 at 5:29
  • $\begingroup$ @LiamKeenan Can you choose some $V$ such that $\mathcal O_X^n|_V\to \mathcal F|_V$ is isomorphic? $\endgroup$ – Born to be proud Dec 10 '17 at 6:24
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    $\begingroup$ I'm sorry for my misleading comments. Quasi-coherent does not a priori imply finitely generated. $\endgroup$ – LPK Dec 10 '17 at 6:40
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Yes, your argument is correct. Quasi-coherence is not necessary. And indeed this is true (and your proof valid) with $X$ an arbitrary ringed space.

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