Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

up vote 6 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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.