  We construct a local Cohen-Macaulay ring $R$ with a prime ideal
$\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander
condition (UAC), but the localization $R_{\mathfrak{p}}$ 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 $\mathfrak{p}\in\spec(R)$ 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.
