p-adic completion




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 include

See also

Last revised on September 20, 2018 at 12:09:27. See the history of this page for a list of all contributions to it.