For a big class of commutative rings
R
R
, every continuous
R
R
-automorphism of
R
[
[
X
1
,
…
,
X
n
]
]
R[[X_1,\ldots ,X_n]]
with the linear part the identity is in the commutator subgroup of
Aut
(
R
[
[
X
1
,
…
,
X
n
]
]
)
\operatorname {Aut}(R[[X_1,\ldots ,X_n]])
. An explicit bound for the number of commutators involved and a
K
K
-theoretic interpretation of this result are provided.