The Stacks project

Lemma 26.7.7. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. The direct sum of an arbitrary collection of quasi-coherent sheaves on $X$ is quasi-coherent. The same holds for colimits.

Proof. Suppose $\mathcal{F}_ i$, $i \in I$ is a collection of quasi-coherent sheaves on $X$. By Lemma 26.7.5 above we can write $\mathcal{F}_ i = \widetilde{M_ i}$ for some $R$-module $M_ i$. Set $M = \bigoplus M_ i$. Consider the sheaf $\widetilde{M}$. For each standard open $D(f)$ we have

\[ \widetilde{M}(D(f)) = M_ f = \left(\bigoplus M_ i\right)_ f = \bigoplus M_{i, f}. \]

Hence we see that the quasi-coherent $\mathcal{O}_ X$-module $\widetilde{M}$ is the direct sum of the sheaves $\mathcal{F}_ i$. A similar argument works for general colimits. $\square$

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