Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen–Macaulay ring R with a prime ideal p∈Spec(R) such that R satisfies the uniform Auslander condition (UAC), but the localization R_p does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal p∈Spec(R) such that R has exactly two non-isomorphic semidualizing modules, but the localization R_p has 2ⁿ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type.