Inert Extensions of Krull Domains

Abstract

Let $A \subseteq B$ be integral domains with $B$ an inert extension of a Krull domain $A$. Let $\mathcal {P}(A)$ be the set of height one primes of $A$, and let $T = { \cap _{p \in \mathcal {P}(A)}}B \otimes {A_p}$. When each ${B_p} = B \otimes {A_p}$ is a UFD, a necessary and sufficient condition for $T$ to be a Krull domain is obtained. If $T$ is a Krull domain and each ${B_p}$ is a UFD, then the divisor class groups of $A$ and $T$ are isomorphic under the natural mapping. These results are applied to $A \subseteq B$ when $B$ is a symmetric algebra over $A$ and when $B$ is locally a polynomial ring over $A$.