nLab p-adic completion



Higher algebra

Higher linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




pp-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 AA an abelian group (or commutative ring) and pp a prime number, the pp-completion of AA is the limit

A p lim n1A/(p nA). 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.

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

Of a homotopy type

(…) (e.g. May-Ponto, 10.2)


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


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

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


Classical accounts:

See also

Last revised on January 22, 2023 at 20:38:03. See the history of this page for a list of all contributions to it.