Inert extensions of Krull domains

Abstract

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