nLab analytic completion of a ring

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

• Aldridge Bousfield, Daniel Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol 304, Springer 1972

• Charles Rezk, Analytic completion (pdf)