= Completed localization of an adic ring
{title2=$A_{\{f\}}=\varprojlim_n(A/I^{n+1})_f$}
For a complete Noetherian <ring> with ideal of definition $I$, completed localization is the completion of the <localization of a ring> $A_f$ in the induced <adic topology>. It gives the sections of the <formal spectrum> over $D(f)$. For $A=k[t][[q]]$, its completed localization at $t$ is $k[t,t^{-1}][[q]]$. Ordinary localization allows a bounded pole order in $t$ across all coefficients; the latter <ring> also contains $\sum_{n\geq0}q^nt^{-n}$, whose pole orders are unbounded.
// Target: ringed-space.bigb
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