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Can we explicitely describe the completion $R$ of $\mathbb{Z}[x]$ with respect to a maximal ideal $\mathfrak{m}\subset \mathbb{Z}[x]$?

$(R,n)$ is a complete regular local ring of dimension two with mixed characteritic. I recently learned that that in this case there is also a structure theorem due to Cohen, which can be found for example in Matsumura's "Commutative ring theory".

If $char(R/n)=p$ one has to look at two cases: a) $p\notin n^2$, b) $p\in n^2$. In the first case $R$ is called unramified and ramified in the other case.

For example in the case (a) $R$ contains a complete DVR $V$ and one has $R\cong V[[x]]$. The case (b) is somehow more complicated.

Now if $\mathfrak{m}=(p,f)$ for some prime $p$ and a polynomial $f$ irreducible mod $p$, can we say for example when case (a) applies to $R$ depending on $p$ and $f$? What is the complete DVR $V$ in this case? Is it alway $\mathbb{Z}_p$?

Or is Cohen's result rather abstract and cannot be made explicit? What about the example $\mathfrak{m}=(p,x)$, do we get $R\cong\mathbb{Z}_p[[x]]$?

ctk
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DonD
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1 Answers1

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Consider the general case $\mathfrak{m}=(p,f)$. Then $n=\mathfrak{m}R$, the maximal ideal of $R$, is also generated by $p$ and $f$. Since $R$ is a regular local ring of dimension two, $\{p,f\}$ is a minimal set of generators for $n$. That is, $p\notin n^2$ (see pp. 25-26).

I presume that you're referring to Theorem 29.7 of Matsumura (p. 228). In fact, the theorem tells us that $R\cong V[[x]]$, where $V$ is a complete $p$-ring with residue field isomorphic to that of $R$. Recall that a $p$-ring is a DVR of characteristic $0$ with uniformiser $p$. Also, note that the residue field of $R$ is the field with $p^d$ elements, where $d=\deg f$.

Now we need the corollary to Theorem 29.2 of Matsumura (p. 225):

A complete $p$-ring is uniquely determined up to isomorphism by its residue field.

So we may take $V$ to be the integral closure of $\mathbb{Z}_p$ in the degree $d$ unramified extension of $\mathbb{Q}_p$. In particular, when $d=1$, we have $R\cong \mathbb{Z}_p[[x]]$.