= Compact Nakayama lemma
{c}
{title2=$M/(p,T)M\text{ finite}\Longrightarrow M\text{ finitely generated over }\Lambda$}
For a compact continuous <module> over the <Iwasawa algebra of a Zp-extension>, lifts of a basis of $M/(p,T)M$ generate $M$. Their generated image is compact. The quotient $Q$ satisfies $Q=(p,T)Q$; in every finite continuous quotient, $(p,T)$ acts nilpotently, since the group action is pro-$p$. Thus all finite quotients of $Q$ vanish, and compactness gives $Q=0$.
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