# Kernels of power surjective maps

Suppose $k$ is an algebraically closed field, and $A$ and $B$ are finitely generated, commutative, graded $k$-algebras. Suppose $\varphi:A\to B$ is a map of $k$-algebras. Notice if $B$ is a domain, then $\ker \varphi$ is prime, and if $\varphi$ is graded, then $\ker\varphi$ is homogeneous, so that certain properties of the map $\varphi:A\to B$ imply ideal-theoretic properties of $\ker\varphi$. Consider the following definition:

A map $\varphi:A\to B$ is called power surjective if for each $b\in B$ there is some $n\in\mathbb{N}$ such that $b^n\in\mathrm{im}\;\varphi$.

I'm interested in describing kernels of power surjective maps. What properties of $\ker\varphi$ are implied by the condition that $\varphi$ is power surjective? That is, what is a non-trivial ideal-theoretic condition $(\star)$ such that the statement

If $\varphi:A\to B$ is power surjective, then $\ker\varphi$ satisfies $(\star)$

is true?

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Hmm. +1 for interesting stuff. But can't you make any homogeneous prime ideal the kernel of such a map $\phi$ given that surjective $\implies$ power surjective? Are you really asking about the properties of the induced mapping between projective varieties? –  Jyrki Lahtonen Jul 3 '14 at 20:38

There won't be any condition, since $\phi : A \to B$ and $A/\mathrm{ker}(\phi) \to B$ have the same image.