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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 2 n 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.