Let
be a complete local (Noetherian) domain such that
. In addition, suppose
contains the rationals,
, and the set of all principal
height-1 prime ideals of
has
the same cardinality as
.
We construct a universally catenary local unique factorization domain
such that the
completion of
is
and such that there exist uncountably many height-1 prime ideals
of
such that
is a field. Furthermore, in
the case where
is a normal
domain, we can make
“close” to excellent in the following sense: the formal fiber at every prime ideal
of
of
height not equal to 1 is geometrically regular, and uncountably many height-1 prime
ideals of
have geometrically regular formal fibers.
Keywords
completions of local rings, excellent rings, unique
factorization domains
