nLab analytic completion of a ring




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

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

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


Last revised on February 5, 2024 at 15:40:23. See the history of this page for a list of all contributions to it.