nLab completion of a ring

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 $R$, which is its topological completion with respect to the adic topology induced by a maximal ideal $I\subset R$ (Sullivan 05, definition 1.3).

The underlying ring $\widehat R_I$ of this formal completion is the limit

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

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

In words, this limit construction says that the elements of $R_I$ are sequences of elements in $R$ which “successively add smaller and smaller elements, as seen by the ideal $I$”. This is as for formal power series rings, which are indeed the archetypical example of formal completions, see example 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 $Spf(R,I)$. Geometrically this is the formal neighbourhood of the spectrum $Spec(R/I)$ inside $Spec(R)$.

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

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

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

More generally:

Example

(completion of Noetherian ring by power series)
For $R$ a Noetherian ring and $I = (a_1, \cdots, a_n)$ a finitely generated ideal, the completion of $R$ at $I$ is isomorphic to the power series ring over $R$ on the $a_i$, in that

$$R\hat{_{I}} \;\simeq\; R [ [ x_1, \cdots, x_n ] ] / \big( x_1 - a_1, \cdots, x_n - a_n \big) \,.$$

(e.g. The Stacks Project Lemma 0316, Buchholtz 08, Sec. 6.4)

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 $(p) \subset \mathbb{Z}$. Similarly the p-adic rational numbers are the completion of the rational numbers $(p)$, and the p-adic complex numbers are the completion of the complex numbers at $(p)$.

Remark

In view of the example one sees that the $p$-adic numbers in 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 $\mathbb{F}_1[x]$ over F1”.

Example

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

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

Example

The phenonemon of example appears for other generalized cohomology theories too, such as complex cobordism (GreenleesMay 97). For complex oriented cohomology theories it says that the formal group assigned by these to $\ast \sslash U(1)$ 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 $\mathfrak{a}\subset A$ of a commutative ring $A$, then the derived functor of $\mathfrak{a}$-adic completion of A-modules forms together with $\mathfrak{a}$-torsion approximation an adjoint modality on the
(∞,1)-category of modules over $A$. See at arithmetic fracturing for chain complexes for details.

As modality in arithmetic cohesion

cohesion in E-∞ arithmetic geometry:

cohesion modality	symbol	interpretation
flat modality	$\flat$	formal completion at
shape modality	$ʃ$	torsion approximation
dR-shape modality	$ʃ_{dR}$	localization away
dR-flat modality	$\flat_{dR}$	adic residual

the differential cohomology hexagon/arithmetic fracture squares:

$$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$$

Related concepts

• analytic completion

• localization of a ring, localization of a commutative ring

• completion of a module

• formal scheme

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

• Wikipedia, Completion (ring theory)

• Brett Bridges, An introduction to ring completions, lecture notes 2011 (pdf)

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

• John Greenlees, Peter May, Derived functors of I-adic completion and local homology, J. Algebra 149 (1992), 438–453 (pdf)

• Leovigildo Alonso, Ana Jeremías, Joseph Lipman, Local Homology and Cohomology on Schemes (arXiv:alg-geom/9503025)

• Marco Porta, Liran Shaul, Amnon Yekutieli, On the Homology of Completion and Torsion (arXiv:1010.4386)

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

• Gunnar Carlsson, Derived completions in stable homotopy theory (arXiv:0707.2585)

Completion for complex cobordism theory is in

• John Greenlees, Peter May, Localization and Completion Theorems for MU-Module Spectra, Annals of Mathematics Second Series, Vol. 146, No. 3 (Nov., 1997), pp. 509-544