I’ve been working on a problem, and come across an issue with capitulation in Dedekind domains. Here is the set up:
Let $D$ be a Dedekind domain, and $K$ its (perfect, but we’re willing to modify assumptions if that significantly changes the answer) field of fractions. Let $L$ be a finite extension of $K$, and $E$ the integral closure of $D$ in $L$.
We say an ideal $I$ in $D$ “capitulates” if it becomes principal in $E$. Let $\text{Cl}(D)$ be the class group of $D$, and we call the subgroup of ideals which capitulate the “capitulation kernel”.
Question: Can the capitulation kernel be infinite?
What we know:
1) We know that the order of the capitulation kernel can be larger than the degree extension $[L:K]$, for example, we can construct quadratic extensions with kernel isomorphic to $\mathbb{Z}/2 \times \mathbb{Z}/2$
2) We know that if the kernel is infinite, then $E$ has an infinitely generated unit group.
3) We know that only the torsion part of $\text{Cl}(D)$ can be in the capitulation kernel, so for this to occur it must be that $\text{Cl}(D)$ has infinitely generated torsion.
4) We know that the following related scenario can happen: for any given abelian group $G$, it is possible to have $D$ be a PID, and $L$ a degree 2 extension of $K$, and $\text{Cl}(E) = G$. (This is thanks to “The Class Group of Dedekind Domains” by Leedham-Green and an earlier paper “Every Abelian Group Is A Class Group” by Claborn.).
5) This question by Pete Clark was interesting, but unfortunately, we weren’t able to make use of it in our question. Two questions about finiteness of ideal classes in abstract number rings
In particular, nothing we have found studies the extensions of Dedekind domains (some study extensions of their localizations, but we’ve been unable to leverage this). Any help, suggestions, or references would be appreciated.
We have a 2nd question motivated by “What we know #2”:
Question: If $D$ has finitely generated unit group, is it possible for $E$ to have infinitely generated unit group?
We have made only a little progress on this problem—we believe we’ve answered in the negative when $L$ is a cyclic extension of $K$.
Thank you for any help you can offer.