= Topologies on integral formal power series
{title2=$(p)\text{-adic and }(p,T)\text{-adic topologies on }\mathbb Z_p[[T]]$}
The <formal power series ring> $R=\mathbb Z_p[[T]]$ is complete for two useful <topologies>. In the $p$-adic topology, convergence means that all coefficient errors are uniformly divisible by arbitrarily large powers of $p$. In the $(p,T)$-adic topology, only finitely many coefficients are controlled at each precision. The latter topology is compact, since $R/(p,T)^n$ is finite and $R$ is its <inverse limit>. The former is finer and is not compact: the series $T^n$ converge to zero in the latter but remain pairwise separated modulo $p$ in the former. <Coleman norm contraction> uses the former; compact interpolation arguments use the latter.
Back to article page
