I am a bit embarrassed to ask this question, but still: assume that I have a finite morphism $\pi:X\to Y$ of affine algebraic varieties over a field (in my case it probably finiteness is a finite morphismtoo strong an assumption, but that shouldn't be relevant)it is true in my situation, so let me assume it), which is an isomorphism over an open subset $U$ of $Y$. Let $Z$ be the complement of $U$. Assume that I am given a regular function $f$ on $X$ which vanishes on the scheme-theoretic preimage $\pi^{-1}(Z)$ of $Z$. Does it come from a function on $Y$?
I am a bit embarrassed to ask this question, but still: assume that I have a morphism $\pi:X\to Y$ of affine algebraic varieties over a field (in my case it is a finite morphism, but that shouldn't be relevant), which is an isomorphism over an open subset $U$ of $Y$. Let $Z$ be the complement of $U$. Assume that I am given a regular function $f$ on $X$ which vanishes on the scheme-theoretic preimage $\pi^{-1}(Z)$ of $Z$. Does it come from a function on $Y$?