The Stacks project

Definition 10.96.2. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. We say $M$ is $I$-adically complete if the map

\[ M \longrightarrow M^\wedge = \mathop{\mathrm{lim}}\nolimits _ n M/I^ nM \]

is an isomorphism1. We say $R$ is $I$-adically complete if $R$ is $I$-adically complete as an $R$-module.

[1] This includes the condition that $\bigcap I^ nM = 0$.

Comments (2)

Comment #7922 by Peng Du on

I suggest to write (0) in the footnote as '0' or '{0}'.

Comment #8173 by on

OK, I changed this but I think sometimes people use to indicate the zero module. This sort of makes sense if you think of as the ideal generated by in the ring . See this commit.

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View Definition 10.96.2 as pdf