Remark 1

Tag 0CWW, permanent. Cite it as [Bourbaki, Tag 0CWW].

Commutative Algebra, chapter IX, § 2. Anneaux de Cohen

Let us retain the preceding hypotheses and notation. We have

a = \sum_{n=0}^{\infty} p^n \tau_k(a_n^{p^{-n}}) \quad (a = (a_n)_{n \in \mathbf{N}} \in \mathbf{W}(k))

by Proposition 7 of § 1, No. 8. Hence (4)

u(a) = \sum_{n=0}^{\infty} p^n \varphi(a_n^{p^{-n}})

for every

a = (a_n)_{n \in \mathbf{N}}

\mathbf{W}(k)

, since

u

is continuous (Theorem 2, b)). By formula (4), the unique Cohen subring of A consists of the elements of the form

\sum_{n=0}^{\infty} p^n s_n

with

s_n \in S

for every integer

n \geq 0

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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