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