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: