Let $Y \rightarrow \mathbb{P}^n$ be a finite scheme and let $Z \rightarrow \mathbb{P}^n$ be a closed subscheme. I read that there is an exact sequence
$0 \rightarrow \mathcal{I}_Z \cdot \mathcal{O}_Y \rightarrow \mathcal{O}_Y \rightarrow \mathcal{O}_{Z \cap Y}\rightarrow 0$.
I can't figure out why this is the case. The description of the inverse image ideal sheaf in Hartshorne is pretty abstract, so I don't how to prove this by working over affine sets.
