Explicit Construction for Cohen’s $p$-ring with imperfect residual field

Asked Jun 19 '23 at 14:30

Active Jun 19 '23 at 15:11

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Let $p>0$ be a prime. Recall that a $p$-ring is a complete DVR $A$ with residual filed $k$ of characteristic $p$ such that $p$ is a uniformizer.

It is known (e.g. Ch 29. of Matsumura’s Commutative Ring Theory) that for any field $k$ of characteristic $p$ there is a unique (up to isomorphism) $p$-ring with residual field $k$.

The ring of Witt vectors $A=W(k)$ offers an explicit construction for perfect $k$, e.g. Serre’s Local Fields.

Is there a similar description when $k$ is imperfect? If so, what is it? Any reference(s)?

edited Jun 19 '23 at 15:11

asked Jun 19 '23 at 14:30

aaa acb

@Flyingpencil I’ve already mentioned Serre’s book in main text of the question. Its discussion of Witt vectors assumes $k$ is perfect. – aaa acb Jun 21 '23 at 01:02

Explicit Construction for Cohen’s $p$-ring with imperfect residual field

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