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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$\begingroup$ 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 $P$? $\endgroup$– prochetCommented Aug 30, 2013 at 19:40
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1 Answer
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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.
answered Aug 30, 2013 at 21:23
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$\begingroup$ And what about the generalisation I asked in my comment? $\endgroup$– prochetCommented Aug 31, 2013 at 14:49
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$\begingroup$ 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. $\endgroup$ Commented Aug 31, 2013 at 21:21