# A concrete meaning of quasi-compact and quasi-separated schemes

This is exercise 5.1.H in Vakil's note 'FOAG':

Show that a scheme $$X$$ is quasi-compact and quasi-separated iff $$X$$ can be covered by a finite number of affine open subsets, any two of which have intersection also covered by a finite number of affine open subsets.

My idea is as follows:

If $$X$$ is quasi-compact and quasi-separated, then $$X$$ can be covered by a finite number of affine open subsets, any two of which have intersection also covered by a finite number of affine open subsets by the previous exercise 5.1.F:

A scheme is quasi-separated iff the intersection of any two affine open subsets is a finite union of affine open subsets.

This is easy and I have solved it. My question is below:

Conversely, we set the scheme $$X$$ can be covered by a finite number of affine open subsets, any two of which have intersection also covered by a finite number of affine open subsets. Let $$X=\bigcup_{i=1}^nU_i$$ where $$U_i$$ are affine open subsets such that any two of which have intersection also covered by a finite number of affine open subsets. So $$X$$ is obviouly quasi-compact. We just need to prove that the intersection of any two affine open subsets is a finite union of affine open subsets by the previous exercise 5.1.F as above.

(*) Take any affine open subset $$Y$$ and if we can prove that $$Y\cap U_i$$ can be written as a finite union of affine open subsets, then the question is finished:

Take $$A,B$$ are two affine open subsets of $$X$$,then $$A\cap B=\bigcup_{i=1}^n((A\cap U_i)\cap(B\cap U_i))$$. Use (*) we know that $$A\cap U_i=\bigcup_{\alpha}C_{\alpha},B\cap U_i=\bigcup_{\beta}D_{\beta}$$ where $$\alpha,\beta$$ are finite and $$C_i,D_i$$ are affine open subsets. Then we find that $$(A\cap U_i)\cap(B\cap U_i)=\left(\bigcup_{\alpha}C_{\alpha}\right)\cap\left(\bigcup_{\beta}D_{\beta}\right)\subset U_i.$$ Since $$U_i$$ is affine, then it is quasi-separated, then $$\left(\bigcup_{\alpha}C_{\alpha}\right)\cap\left(\bigcup_{\beta}D_{\beta}\right)=\bigcup_{\alpha,\beta}(C_{\alpha}\cap D_{\beta})$$ is a finite union of affine open subsets, then so is $$A\cap B$$.

So the question is: how to prove (*)? or how to prove the exercise without using this?

I’m sorry, this is a trivial problem. Just need to prove that $$Y\cap U_i$$ is quasi-compact. This is very easy.