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Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?

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More generally, if we have a polynomial $P\in k[\epsilon][t]$ which reduces to $t^{N}$ for an integer $N$, can we describe the completion of the ring $k[\epsilon][t]$ with respect to $P$? – prochet Aug 30 '13 at 19:40
up vote 7 down vote accepted

If the characteristic of $k$ is not $2$, $(t^2+\epsilon)=(t+\epsilon/2)^2$. So write $t'=t+\epsilon/2$. Them we are looking at the $t'^2$-adic completion of $k[\epsilon][[t']]$.

If the characteristic is $2$, then $(t^2+\epsilon)^2=t^4$. So this is just the $t$-adic completion.

Alternately, you can observe that a sequence is Cauchy with respect to this ideal if and only if it is Cauchy with respect to $t$, and the same for sequences converging to $0$, so the completions ae the same.

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And what about the generalisation I asked in my comment? – prochet Aug 31 '13 at 14:49
The same techniques apply. $(t^N+\epsilon f) = (t+\epsilon f/N)^N$ or in characteristic $p$ dividing $N$ we have $(t^N+\epsilon f)^p = t^{Np}$. Or the Cauchy sequence thing. – Will Sawin Aug 31 '13 at 21:21

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