nLab p-adic completion

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)

Related concepts

• p-adic integers

• completion of a group

• completion of a ring, analytic completion of a ring

• p-localization

• p-adic homotopy theory

• rationalization, rational homotopy theory

• fracture theorem

• nilpotent homotopy type, finite homotopy type

• nilpotent completion of spectra

• L-complete module

References

Classical accounts:

• Dennis Sullivan, Geometric topology: localization, periodicity and Galois symmetry, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface

  by Andrew Ranicki (pdf)

• Aldridge Bousfield, Daniel Kan, Localization and completion in homotopy theory, Bull. Amer. Math. Soc. 77 6 (1971) 1006-1010 [doi:10.1090/S0002-9904-1971-12837-9, pdf]

• Aldridge Bousfield, Daniel Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, 304 Springer (1972) [doi:10.1007/978-3-540-38117-4]

• Peter May, Kate Ponto, chapters 7 and 8 of More concise algebraic topology: Localization, completion, and model categories (pdf)

See also

• Doug Ravenel, chapter 2, def. 2.1.14 Complex cobordism and stable homotopy groups of spheres

• John Rognes, section 4.6 of The Adams spectral sequence, 2012 (pdf)