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completion of a ring

Context

Algebra

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Formal geometry

Contents

Ideas

General

Quite generally, the completion of a ring is a completion of a topological ring to a complete topological ring, when possible, for instance of a normed ring to a Banach ring.

Formal/adic completion and formal neighbourhoods

A special case of ring completion is the formal completion or adic completion of a commutative ring RR, which is its topological completion with respect to the adic topology induced by a maximal ideal IRI\subset R (Sullivan 05, definition 1.3).

The underlying ring R^ I\widehat R_I of this formal completion is the limit

R^ Ilim(R/I n) \widehat R_I \coloneqq \underset{\leftarrow}{\lim} (R/I^n)

(formed in the category CRing of commutative rings) of the quotients of RR by all the powers of this ideal, II (Sullivan 05, proposition 1.13). Notice that this may be considered purely algebraically.

In words, this limit construction says that the elements of R IR_I are sequences of elements in RR which “successively add smaller and smaller elements, as seen by the ideal II”. This is as for formal power series rings, which are indeed the archetypical example of formal completions, see example 1 below.

Generally, the dual geometric meaning of formal ring completion is in formal geometry: the proper geometric spectrum of a formally completed ring is known as a formal spectrum Spf(R,I)Spf(R,I). Geometrically this is the formal neighbourhood of the spectrum Spec(R/I)Spec(R/I) inside Spec(R)Spec(R).

Spec(R/I)Spf(R^ I)Spec(R). Spec(R/I) \hookrightarrow Spf(\widehat R_I) \hookrightarrow Spec(R) \,.

Derived completion functor

The derived functor of adic completion was originally discussed in (Greenless-May 92 (“Greenlees-May duality”). For discussion of its relation to derived torsion subgroup functor see (Porta-Shaul-Yekutieli 10) and see at fracture theorem – Arithmetic fracturing for chain complexes.

Examples

The archetypical example which most clearly exhibits the geometric meaning of formal completions of rings is the following

Example

For RR any ring and R[x]R[x] the polynomial ring with coefficients in RR, then the formal completion of R[x]R[x] at the ideal (x)(x) generated by the free generator xx is the ring of formal power series R[[x]]R[ [x] ].

If RR is a field, then geometrically Spec(R[x])Spec(R[x]) is the affine line in algebraic geometry/arithmetic geometry over RR, while Spf(R[[x]])Spf(R[ [x] ]) is the formal disk inside the affine line around the origin.

The key class of example of completions in non-archimedean analytic geometry is the following.

Example

The p-adic integers are the completion of the ring of integers at the prime ideal (p)(p) \subset \mathbb{Z}. Similarly the p-adic rational numbers are the completion of the rational numbers (p)(p), and the p-adic complex numbers are the completion of the complex numbers at (p)(p).

Remark

In view of the example 1 one sees that the pp-adic numbers in 2 are in fact analogous to formal power series rings, hence that they behave like function rings on formal disks in some kind of geometry.

This analogy is part of what is known as the function field analogy, which says that the ring of integers \mathbb{Z} behaves like the would-be “ring of polynomials 𝔽 1[x]\mathbb{F}_1[x] over F1”.

Example

The Atiyah-Segal completion theorem states that for GG a topological group and XX a G-space, then the topological K-theory ring K(X//G)K(X//G) of the homotopy quotient X//GX//G (of the Borel construction) is the completion of the GG-equivariant K-theory ring of XX.

So this says that the “very naive” equivariant K-theory embodied by K(X//G)K(X//G) is an infinitesimal approximation to the genuine equivariant K-theory.

Example

The phenonemon of example 3 appears for other generalized cohomology theories, too. For complex oriented cohomology theories it says that the formal group assigned by these to *//U(1)\ast//U(1) is to be thought of as the formal completion of a more globally defined something. One case where this “something” has been understood in some detail is that where the cohomology theory is elliptic cohomology. In that case the analog of equivariant K-theory is equivariant elliptic cohomology, see there for more details.

Properties

Relation of formal completion to torsion approximation

For suitable ideals 𝔞A\mathfrak{a}\subset A of a commutative ring AA, then the derived functor of 𝔞\mathfrak{a}-adic completion of A-modules forms together with 𝔞\mathfrak{a}-torsion approximation an adjoint modality on the
(∞,1)-category of modules over AA. See at arithmetic fracturing for chain complexes for details.

As modality in arithmetic cohesion

cohesion in E-∞ arithmetic geometry:

cohesion modalitysymbolinterpretation
flat modality\flatformal completion at
shape modalityʃʃtorsion approximation
dR-shape modalityʃ dRʃ_{dR}localization away
dR-flat modality dR\flat_{dR}adic residual

the differential cohomology hexagon/arithmetic fracture squares:

localizationawayfrom𝔞 𝔞adicresidual Π 𝔞dR 𝔞X X Π 𝔞 𝔞dRX formalcompletionat𝔞 𝔞torsionapproximation, \array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,

References

A classical account is in section 1 of

  • 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)

Brief surveys include

Discussion of the derived functor of adic completion (“Greenless-May duality”) is in

for more on this see also at fracture theorem – Arithmetic fracturing for chain complexes

Discussion of interrelation between completion and etale morphisms is in

  • Leovigildo Alonso, Ana Jeremias, Marta Perez, Local structure theorems for smooth maps of formal schemes (arXiv:math/0605115)

Discussion of formal completion of (infinity,1)-modules in terms of totalization of Amitsur complexes is in

Revised on February 17, 2017 13:01:58 by Yohannes D. A.? (137.146.138.0)