Let
R
R
be a Noetherian domain and
P
P
a prime ideal of
R
R
. Then
R
p
[
[
X
1
,
…
,
X
n
]
]
{R_p}[[{X_1}, \ldots ,{X_n}]]
has a maximal chain of prime ideals of length
r
r
if and only if
R
[
[
X
1
,
…
,
X
n
]
]
(
P
,
X
1
,
…
,
X
n
)
R{[[{X_1}, \ldots ,{X_n}]]_{(P,{X_1}, \ldots ,{X_n})}}
does, if and only if
R
[
[
X
1
,
…
,
X
n
]
]
(
P
,
X
1
,
…
,
X
n
)
R{[[{X_1}, \ldots ,{X_n}]]_{(P,{X_1}, \ldots ,{X_n})}}
does.