The Picard group of an integer-valued polynomial ring
Let \(A\) be a one dimensional noetherian domain with finite residue fields. The author study the Picard group of the ring \(B=\{f \in K[X]|\) \(f(a) \in A\) for all \(a \in A\}\) of integer valued polynomials over \(A\). First he reduces the problem to the local ring case, then he proves that \(\text{Pic}(B)\) is isomorphic to the class group \({\mathcal C}(A,{\mathcal P}(A))/{\mathcal P}(A)\) of locally constant functions from \(A\) to the group \({\mathcal P}(A)\) of principal fractional ideals of \(A\) if and only if the \(m\)-adic completion \(A^*\) is a domain. Special cases of this result were known, for example when the ring \(A\) is a countable set.
scientific article; zbMATH DE number 63902