We construct a local Cohen-Macaulay ring $R$ with a prime ideal $mathfrak{p}inspec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{mathfrak{p}}$ does not satisfy Auslanders condition (AC). Given any positive integer $n$, we also construct a local Cohen-Macaulay ring $R$ with a prime ideal $mathfrak{p}inspec(R)$ such that $R$ has exactly two non-isomorphic semidualizing modules, but the localization $R_{mathfrak{p}}$ 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.
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