= Adic completion of a module
{title2=$\widehat M=\varprojlim_n M/I^nM$}
The adic completion is the <inverse limit> of the <quotient modules> $M/I^nM$. An element is a compatible family of residue classes at every order. For $R=k[t]$ and $M=V\otimes_k k[t]$, this gives the <formal power series module> $V[[t]]$. The canonical map from $M$ need not be injective in general: its <kernel> is $\bigcap_n I^nM$. It is injective for this polynomial example because a nonzero polynomial has finite degree. Successive changes of coordinates in <formal rigidity from vanishing second Hochschild cohomology> converge in this completion because the order-$r$ change is the identity modulo $t^r$.
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