The $h$-adic topology

1.1 Let $A$ be a ring. Let $𝔞$ be an ideal in $A$. We view the powers ${𝔞}^{k}$ of the ideal $𝔞$ as a basis of neighbourhoods in $A$ containing $0$. There is a unique topology on $A$ such that the ring operations are continuous with a basis given by the sets $a+{𝔞}^{k}$. This is the $𝔞$-adic topology. If ${\cap }_{k}{𝔞}^{k}=\left(0\right)$ then this topology is Hausdorff.

1.2 Let $M$ be an $A$-module. We can transfer the $𝔞$-adic topology on $A$ to a topology on $M$. We view the sets ${N}_{k}={𝔞}^{k}M$ as a basis of neighbourhoods in $M$ containing $0$. An element $m\in M$ is an element of ${N}_{k}$ if . As above, there is a unique topology on $M$ such that the module operations are continuous with basis given by the sets $m+{𝔞}^{k}M$, where $m\in M$. This is the ${𝔞}^{k}$-adic topology on $M$.

1.3 Define a map $d:M × M\to ℝ$ by where $e$ is a real number $e>1$ and $v\left(x\right)$ is the largest integer $k$ such that $x\in {𝔞}^{k}M$. If the $𝔞$-adic toplogy on $M$ is Hausdorff the $d$ is a metric on $M$ which generates the $𝔞$-adic topology.

1.4 If $A$ is a local ring then it is natural to take $I=𝔪$ where $𝔪$ is the unique maximal ideal in $A$. If $k$ is a field and $h$ is an indeterminate then the ring of formal power series in $h$, $k\left[\left[h\right]\right]$, is a local ring with unique maximal ideal $𝔪=\left(h\right)$ generated by $h$. In this case the $𝔪$-adic topology on a $k\left[\left[h\right]\right]$-module $M$ is called the $h$-adic topology on $M$.

1.5 Let $A$ be a ring and $𝔞$ be an ideal of $A$. Let $M$ be an $A$-module. A sequence of elements $\left\{{b}_{n}\right\}$ in $M$ is a Cauchy sequence in the $𝔞$-adic topology if for every positive integer $k>0$ there exists a positive integer $N$ such that A sequence $\left\{{b}_{n}\right\}$ of elements in $M$ converges to $b\in M$ if for every positive integer $k>0$ there exists a positive integer $N$ such that The module $M$ is complete in the $𝔞$-adic topology if every Cauchy sequenc in $M$ converges. A ring $A$ is complete in the $𝔞$-adic topology if when viewed as an $A$-module it is completd in the $𝔞$-adic topology. If the $𝔞$-adic topology is Hausdorff then this definition of completeness is the same as the ordinary defingion of completenes when we view that $M$ is a metric space as in (1.1).

1.6 Two Cauchy sequences $P=\left\{{p}_{n}\right\}$ and $Q=\left\{{q}_{n}\right\}$ in $M$ are equivalent if $\left\{{p}_{n}-{q}_{n}\right\}$ converges to $0$ in the $𝔞$-adic topology, i.e., The set of all equivalece classes of Caucy sequences in $M$ is the completion $\stackrel{ˆ}{M}$ of $M$

1.7 The completion $\stackrel{ˆ}{M}$ is an $\stackrel{ˆ}{A}$ module with operations given by $P+Q = pn+qn ,and, an P = anpn ,$ where $P=\left\{{p}_{n}\right\}$ and $Q=\left\{{q}_{n}\right\}$ are Cauchy sequences with elements in $M$ and $\left\{{a}_{n}\right\}$ is a Cauchy sequence of elements in $A$.

1.8 Define a map $\phi :M\to \stackrel{ˆ}{M}$ by $φ(b)= [ ( b,b,b,… ) ],$ i.e., $\phi \left(b\right)$ is the equivalence class of the sequence $\left\{{b}_{n}\right\}$ such that ${b}_{n}=b$ for all $n$. This map has kernel ${\cap }_{k}{𝔞}^{k}M.$ The map $\phi$ is injective if $M$ is Haussdorff in the $𝔞$-adic topology.

1.9 Define a basis ${N}_{k}$ of neigbourhoods of $0$ in the completion $\stackrel{ˆ}{M}$ by: The collection of sets $P+{N}_{k}$ where $P\in \stackrel{ˆ}{M}$ is a basis for a topology on $\stackrel{ˆ}{M}$. The module operations and the map $\phi$ are continuous.

1.10 Let $k$ be a field. Then $k\left[\left[h\right]\right]$ is a local ring with maximal ideal $𝔪=\left(h\right)$ generated by the element $h$. In this case the $𝔪$-adic topology is called the $h$-adic toplogy. Let $M$ be a $k\left[\left[h\right]\right]$-module. Then a sequence of elements $\left\{{b}_{n}\right\}$ in $M$ is a Cauchy sequence if for every positive integer $k>0$ there exists a postive integer $N$ such that i.e., ${b}_{n}-{b}_{m}$ is "divisible" by ${h}^{k}$ for all $m,n>N$. A sequence $\left\{{b}_{n}\right\}$ of elements in $k\left[\left[h\right]\right]$ converges to $b\in M$ if for every positive integer $k>0$ there exists a positive integer $N$ such that The module $M$ is complete in the $h$-adic topology if every Cauchy sequence in $M$ converges.

1.11 As in (1.2) we can define the completion of a $k\left[\left[h\right]\right]$-module $M$ in the $h$-adic topology. IF $A$ is an algebra over a field $k$ then $A{\otimes }_{k}k\left[\left[h\right]\right]$ is a $k\left[\left[h\right]\right]$-module in the $h$-adic topology and the completion of $A{\otimes }_{k}k\left[\left[h\right]\right]$ is $A\left[\left[h\right]\right]$, the ring of formal power series in $h$ with coefficients in $A$. The ring $A\left[\left[h\right]\right]$ is, in general, larger than $A{\otimes }_{k}k\left[\left[h\right]\right]$.

1.12 If $M$ is a complete $k\left[\left[h\right]\right]$-module in the $h$-adic toplogy then for each element $x={\sum }_{j\ge 0}{x}_{j}{h}^{j}\in M$ the element $ehx= ∑ k≥0 (hx)k k! = 1+x0h+ (x0h+2x1) ( h2 2 )+ ( x03 +3 ( x0x1+x1x0 ) +6x2 ) ( h3 3! )+⋯$ is a well defined element of $M$.

1.13 A $k\left[\left[h\right]\right]$-module $M$ is topologically free if $M/{h}^{k}M$ is a free $k\left[\left[h\right]\right]/\left({h}^{k}\right)$-module for all positive integers $k>0$.

References

The following books have discussions of the $h$-adic toplogy and completions. The definitions of completion for a metric space are found in Rudin's elementary analysis book Chapt. 3 Exercise 23--24.

[AM] M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley 1969.

[ZS] O. Zariski and P. Samuel, Commutative algebra, Vol. II Van-Nostrand 1960.