Closed subscheme of an affine scheme.

Question

This is a portion of the exercise 2.3.11 (b) of Hartshorne. Suppose $Y$ is a closed subscheme of $X=\operatorname{Spec}A$. I would like to show that there are $f_{i}\in A$ so $D(f_{i})$ cover $X$ and each $D(f_{i})\cap Y$ is an affine open subset of $Y$. I'm not sure how to show this claim (which is indicated as a "hint"), although I think this won't be that hard. Any helps will be appreciated!

Answer 1

In some sense the trick he uses in proving (3.2) is the prototype for how every argument of this sort works. I'll try to get you started -- I'm not really saying anything interesting here.

Pick a point $P \in Y$ and some affine open neighborhood $U$ of $P$ in $Y$. You can find an $f \in A$ such that $P \in D(f) \cap Y \subseteq U$. Argue that $D(f) \cap Y$ is a distinguished open affine in $U$, and that it only takes a finite number of subsets of this form to cover $Y$.