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Let $X$ be a scheme covered by a finite number of affine open subsets $U_i$ such that for any $U_i, U_j$, the $U_i\cap U_j$ is a union of finite number of affine open subsets $W^{(i,j)}_h$. Then for any affine open subsets $U, V$, the $U\cap V$ is a union of a finite number of affine open subsets. This is essentially Vakil's note 6.1.H(p142).

It would be very appreciated if you give an elementary proof.(I knew the definition of schemes only a week ago. All I know is before 6.1.H.) Or any reference?

I want to show that projective schemes are quasi-separated. I know it is true if the above is true.

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I see you already found this thread. Just thought to link it here. –  Gregor Bruns Nov 25 '12 at 11:55
    
Thank you. I didn't know how to link. –  Tom Nov 25 '12 at 12:05
    
But I myself was not satisfied with that answer. It does not mean that it is bad. I cannot judge while it might be enough for advanced learners. –  Tom Nov 25 '12 at 12:17
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Dear Tom, If you want to show projective schemes are quasi-separated just as an exercise because you are learning the subject, why not invest more time in trying to solve the problem yourself instead of asking someone to solve it for you here; Vakil's notes are well-thought out and structured, and if he asks you to solve it at this point, he has given you enought tools to do so. On the other hand, if you need to know that projective schemes are quasi-separated as an ingredient in some other piece of work, than there are many references, e.g. in Hartshorne it is proved that they are ... –  Matt E Nov 25 '12 at 13:12
    
... separated, which implies that they are quasi-separated. Regards, –  Matt E Nov 25 '12 at 13:13

1 Answer 1

Here is a hint: Skip forward in the notes to proposition 6.3.1. There it is proved that you can cover the intersection of two open affines by affine opens that are distinguished (that is $D(\cdot)$) in both of them.

Together with the observation that $U_i\cap U_j$ is quasicompact, I found that ingredient very useful in proving 6.1.H myself some time ago. Maybe try to reduce to the case that $X$ is covered by only two affines.

I would suggest you give the exercise another try. If you get too frustrated, you can still ask again for a solution or another hint.

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Thank you very much for giving me 6.3.1 kindly. I jumped into the proposition and understood its proof. It is helpful. But I haven’t completed the solution yet. Possibly I am misunderstanding something. This is my way of thinking: (1) Let $U_1,U_2$ be affine open sets of a scheme $X$ such that $X=U_1\cup U_2$ with other affine open sets $V_1,V_2$ having $U_1\cap U_2=V_1\cup V_2$. –  Tom Nov 26 '12 at 1:31
    
(2) Let $W$ be an affine open set of $X$. It suffices to show that $U_1\cap W$ is a union of finitely-many affine open sets. (3) There exist (infinitely-many) affine open sets $T_j$ with $U_1\cap W=\cup_j T_j.$ The $T_j$ can be distinguished open sets of both of $U_1$ and $W$ (by 6.3.1). (4) Question: How can we delete all $T_j$ but finitely-many? How do we use $W_1, W_2$? –  Tom Nov 26 '12 at 1:31
    
Sorry in (4), $W_1,W_2$ should be replaced by $V_1,V_2$. In the post they are $W^{(i,j)}_h.$ –  Tom Nov 26 '12 at 1:37

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