nLab
p-adic completion

Contents

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

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)

Properties

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

Examples

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)

References

Classical accounts include

See also

Last revised on November 4, 2020 at 09:24:14. See the history of this page for a list of all contributions to it.