Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $f : A \to B$ be a homomorphism of finitely generated $k$-algebras, where $k$ is a field. Let $J_A$ and $J_B$ denote the conductor ideals of $A$ and $B$ respectively for the corresponding normalizations in the quotient fields (assume that $A, B$ are domains). Is it true that $f(J_A) \subseteq J_B$ ? What is the relationship between $f^{-1}(J_B)$ and $J_A$ ?

share|improve this question
add comment

1 Answer 1

There is no strong relations between $J_A$ and $J_B$ if you don't have some condition on $A\to B$.

First, it is not true that $f(J_A)\subseteq J_B$: just take $A$ integrally closed and $B$ not integrally closed. Then $1\in J_A$ but $1=f(1)\notin J_B$.

Second, suppose $B$ is the integral closure of $A$ and $B\ne A$. Then $J_B=B$ and $J_B$ is not contained in $f(J_A)B$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.