Contents

Contents

Idea

For $A$ a commutative ring and $p$ a prime number, then there is the topological completion at $(p)$, the p-completion $A_p^\wedge \coloneqq \underset{\leftarrow}{\lim}_n A/(p^n A)$. The analytic completion is instead the quotient

$A[ [ x ] ]/(x-p)A[ [ x ] ] \,.$

For $A= \mathbb{Z}$ the integers both constructions agree, up to isomorphism, and yield the p-adic integers. In general though they are different.

References

Last revised on July 21, 2014 at 06:22:23. See the history of this page for a list of all contributions to it.