Is there a notion of inverse image for schemes?

Question

Let $f: X \to Y$ be a morphism of schemes, and let $Z$ be a subscheme of $Y$. Is there any reasonable way to talk about an "inverse image" of $Z$?

Answer 1

The image of the natural morphism $$ f^*I_Z \to f^*\mathcal{O}_Y \cong \mathcal{O}_X $$ is an ideal in $\mathcal{O}_X$. The corresponding subscheme of $X$ is a natural candidate for the inverse image of $Z$.