Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

$p$-completion is to p-adic homotopy theory as rationalization is to rational homotopy theory.

For more see at formal completion.

### Of abelian groups, rings and modules

For $A$ an abelian group (or commutative ring) and $p$ a prime number, the $p$-completion of $A$ is the limit

$A_p^\wedge \coloneqq \underset{\leftarrow}{\lim}_{n \geq 1} A/(p^n A) \,.$

(e.g. May Ponto, 10.1.1) For more see at formal completion.

$A$ is called $p$-complete if the canonical homomorphism $A \to A_p^\wedge$ is an isomorphism.

### Of a homotopy type

(…) (e.g. May-Ponto, 10.2)

## Properties

The fracture theorem says that under mild conditions a (stable) homotopy type decomposes into its rationalization and its $p$-completions.

## Examples

For $A = \mathbb{Z}$ the integers, the $p$-completion is the p-adic integers. (Notice that here traditionally one writes $\mathbb{Z}_p = \mathbb{Z}_p^\wedge$.)

More generally, if $A$ is finitely generated, then $A_p^\wedge \simeq A\otimes \mathbb{Z}_p$. (e.g. May Ponto, p. 154)

## References

Classical accounts include