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Exercise 6.1.G of Ravi Vakil's notes asks to prove that all affine schemes are quasi-separated, where quasi-separated schemes are defined as schemes where the intersection of any two quasi-compact open subsets is quasi-compact, or equivalently the "intersection of any two affine open subsets is a finite union of affine open subsets."

Can someone give a hint or solution?

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up vote 7 down vote accepted

The qc open subsets of $\mathrm{Spec}(A)$ have the form $\cup_i D(f_i)$ with finitely many $f_i \in A$. If we intersect two such sets, we optain $\cup_i D(f_i) \cap \cup_j D(g_j) = \cup_{ij} D(f_i g_j)$, which is a finite union of affine schemes, and therefore qc.

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