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This question maybe finds the answer in the definition. Let $X$ be a scheme with not hypothesis on it (not noetherian, not affine,... ). Can I find a scheme that has a not numerable affine covering? For example, if a take a set of rings $\{R_i\}_{i \in I}$ such that $|I|=|\mathbb{R}|$, is it true that $\bigsqcup_{i \in I} \mathrm{Spec}(R_i)$ is a scheme?

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    $\begingroup$ Dear @Servaes: If $I$ is infinite, it is not true that $\sqcup\text{Spec}(R_i) = \text{Spec}(\prod R_i)$. The right hand side is quasi-compact but the left hand side is not. $\endgroup$ – Bruno Joyal Mar 25 '14 at 17:20
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Yes, $X=\bigsqcup \text{Spec } R_i$ is a scheme.

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