Given a complete local (Noetherian) ring
T
T
, we find necessary and sufficient conditions on
T
T
such that there exists a local domain
A
A
with
|
A
|
>
|
T
|
|A| > |T|
and
A
^
=
T
\widehat {A} = T
, where
A
^
\widehat {A}
denotes the completion of
A
A
with respect to its maximal ideal. We then find necessary and sufficient conditions on
T
T
such that there exists a domain
A
A
with
A
^
=
T
\widehat {A} = T
and
|
S
p
e
c
(
A
)
|
>
|
S
p
e
c
(
T
)
|
|\mathrm {Spec}(A)| > |\mathrm {Spec}(T)|
. Finally, we use “partial completions” to create local rings
A
A
with
A
^
=
T
\widehat {A} = T
such that
S
p
e
c
(
A
)
\mathrm {Spec}(A)
has varying cardinality in different varieties.