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adic completion
The ur-example: p-adic integers.
Idea: for schemes, completion at a point is a formal neighborhood.
Completion wrt a filtration
For \((M, {\operatorname{Fil}}^*)\) a (descending) filetered module, one forms the completion as
\begin{align*}
M = {\operatorname{Fil}}^0 M \supseteq {\operatorname{Fil}}^1 M \supseteq \cdots \implies \widehat{M} \coloneqq M{ {}_{ \widehat{{\operatorname{Fil}}^*} } } \coloneqq\varprojlim_{n}\, {M \over {\operatorname{Fil}}^n M}
\end{align*}
If \({\operatorname{Fil}}^*\) is a terminating filtration, the result is a topological module. # adic completion 
So \begin{align*} R{ {}_{ \widehat{I} } } = \varprojlim_n R/I^n \end{align*}
Usually involves localizing first: 
Results
- Hensel’s Lemma applies to complete rings.
- Coincides with Cauchy completion when \(R\) has a metric induced by a nonarchimedean absolute value.
Examples
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\({\mathbf{Z}}{ {}_{ \widehat{\left\langle{p}\right\rangle} } } = {\mathbf{Z}}{ {}^{ \wedge }_{p} }\) is the p-adic integers.
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\(k[x_1,\cdots, x_n]{ {}_{ \widehat{\left\langle{x_1,\cdots, x_n}\right\rangle} } } = k{\left[\left[ x_1, \cdots, x_n \right]\right] }\) are multivariate formal power series. # Exercises
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Show that the completion of a Noetherian ring \(R\) is flat as an \(R{\hbox{-}}\)module.
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Show that completion can be computed by extension of scalars as \begin{align*} M { {}_{ \widehat{I} } } \cong M \otimes_R R{ {}_{ \widehat{I} } } \end{align*}
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Use this to show \((R/J){ {}_{ \widehat{I} } } \cong R{ {}_{ \widehat{I} } }/J{ {}_{ \widehat{I} } }\).
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Relation to 