Does every affine open of a closed subscheme come from an affine open of the ambient scheme?

Question

Let $(\iota,\iota^{\#}):(Y,\mathcal{O}_Y)\to (X,\mathcal{O}_X)$ be a closed immersion of schemes. Is it true that for every affine open $V\subseteq Y$, there exists an affine open $U\subseteq X$ with $\iota^{-1}U=V$? Of course there exists an open set $\tilde{U}\subseteq X$ such that $V=\iota^{-1}\tilde{U}$, and then as $V$ is quasi-compact we may assume that $\tilde{U}$ is a finite union of affines. But can we always reduce it to just being affine?

Answer 1

I don't think so. Every affine open in $\Bbb P^2_k$ is of the form $D(f)$ for some homogeneous $f\in k[x_0,x_1,x_2]$. On the other hand, an elliptic curve $Y\subset \Bbb P^2_k$ with identity $O$ and a point $P$ of infinite order, $Y\setminus\{O,P\}$ is affine but cannot be of the form $D(f)\cap Y$: if so, this would mean that $V(f)\cap Y$ is a divisor supported at $\{0,P\}$ contradicting the fact that $P$ is of infinite order.

It's much better to take an affine open of $X$ and produce an affine open of $Y$ than the other way around.