# Artinian local algebras over a complete local noetherian ring.

So let $\Lambda$ be a complete local noetherian ring.

The author claims that $\Lambda[t]/(t^i)$ is an Artinian local $\Lambda$-algebra with the same residue field as $\Lambda$.

I don't see this. For example, of $\Lambda$ is a complete DVR with uniformizer $\pi$, and $i = 2$, then the descending sequence $(\pi,t)^n$ doesn't seem to stabilize. The first few terms are:

$(\pi,t) \supsetneq (\pi^2,\pi t) \supsetneq (\pi^3,\pi^2t)\supsetneq\cdots$

which obviously doesn't stabilize...or am I missing something?

This is from the very beginning of chapter 2.1 in:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.132.2930

thanks

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## 1 Answer

Note that $\Lambda[t]/(t^i)$ is a free $\Lambda$ module spanned by $1,\ldots,t^{i-1}$, so it is not Artinian unless $\Lambda$ is. Thus the example in your reference is incorrect. In the notation of your reference, this ring is in $\widehat{\mathcal C}_{\Lambda}$ rather than $\mathcal C_{\Lambda}$, unless itself $\Lambda$ is Artinian.

Presumably this doesn't affect anything that follows, and you may do well to bear in mind that the author (now an accomplished and highly respected expert in arithmetic geometry) wrote this as part of his master's thesis, and he explicitly states in the link you give that he hasn't tried to correct it since he first wrote it (and warns that it contains "foolish mistakes").

Incidentally, there are many additional places that you can read deformation theoretic ideas, and learn the roles of the categories $\mathcal C_{\Lambda}$ and $\widehat{\mathcal C}_{\Lambda}$.

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