How to prove $\Delta : X \to X \times_Y X$ is quasi-compact for $X$ noetherian?

Question

Let $X$ be a noetherian scheme, $f: X \to Y$ consider the fibred product $X \times_Y X$. I would like to know how to show the diagonal morphism $\Delta : X \to X \times_Y X$ is quasi-compact. Thank you.

Answer 1

To check that $\Delta : X \to X\times_Y X$ is quasicompact, it is enough to show that the intersection of any two open affine subsets $U$ and $V$ of $X$ is quasicompact. Can you show this? (Can you show that any open subset of $\text{Spec }A$ is quasicompact if $A$ is a Noetherian ring?)