In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a completion is any of several related
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s on
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
and
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
that result in complete
topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps:
R \times R \to R
where R \times R carries the product topology. That means R is an additive ...
s and
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
. Completion is similar to
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
, and together they are among the most basic tools in analysing
commutative rings. Complete commutative rings have a simpler structure than general ones, and
Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to ...
applies to them. In
algebraic geometry, a completion of a ring of functions ''R'' on a space ''X'' concentrates on a formal neighborhood of a point of ''X'': heuristically, this is a neighborhood so small that ''all'' Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to
completion of a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
with
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
s, and agrees with it in the case when ''R'' has a metric given by a
non-Archimedean absolute value.
General construction
Suppose that ''E'' is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
with a descending
filtration
:
of subgroups. One then defines the completion (with respect to the filtration) as the
inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
:
:
This is again an abelian group. Usually ''E'' is an ''additive'' abelian group. If ''E'' has additional algebraic structure compatible with the filtration, for instance ''E'' is a
filtered ring In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.
A filtered algebra over the field k is an alge ...
, a filtered
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
, or a filtered
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, then its completion is again an object with the same structure that is complete in the topology determined by the filtration. This construction may be applied both to
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
and
noncommutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s. As may be expected, when the intersection of the
equals zero, this produces a complete
topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps:
R \times R \to R
where R \times R carries the product topology. That means R is an additive ...
.
Krull topology
In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, the filtration on a
commutative ring ''R'' by the powers of a proper
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
''I'' determines the Krull (after
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
) or ''I''-
adic topology
In commutative algebra, the mathematical study of commutative rings, adic topologies are a family of topologies on elements of a module, generalizing the -adic topologies on the integers.
Definition
Let be a commutative ring and an -module. T ...
on ''R''. The case of a
''maximal'' ideal is especially important, for example the distinguished maximal ideal of a
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' suc ...
. The
basis of open neighbourhoods
In mathematics, a base (or basis) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open ...
of 0 in ''R'' is given by the powers ''I''
''n'', which are ''nested'' and form a descending filtration on ''R'':
:
(Open neighborhoods of any ''r'' ∈ ''R'' are given by cosets ''r'' + ''I''
''n''.) The completion is the
inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
of the
factor ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...
s,
:
pronounced "R I hat". The kernel of the canonical map from the ring to its completion is the intersection of the powers of ''I''. Thus is injective if and only if this intersection reduces to the zero element of the ring; by the
Krull intersection theorem In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
, this is the case for any commutative
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
which is an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
or a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
.
There is a related topology on ''R''-modules, also called Krull or ''I''-
adic topology
In commutative algebra, the mathematical study of commutative rings, adic topologies are a family of topologies on elements of a module, generalizing the -adic topologies on the integers.
Definition
Let be a commutative ring and an -module. T ...
. A basis of open neighborhoods of a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
''M'' is given by the sets of the form
:
The completion of an ''R''-module ''M'' is the inverse limit of the quotients
:
This procedure converts any module over ''R'' into a complete
topological module In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
Examples
A topological vector space is a topological module over a topological field.
An abelian topological g ...
over
.
Examples
* The ring of
''p''-adic integers is obtained by completing the ring
of integers at the ideal (''p'').
* Let ''R'' = ''K''
1,...,''x''''n''">'x''1,...,''x''''n''be the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
in ''n'' variables over a field ''K'' and
be the maximal ideal generated by the variables. Then the completion
is the ring ''K''
1,...,''x''''n''">''x''1,...,''x''''n'' of
formal power series in ''n'' variables over ''K''.
* Given a noetherian ring
and an ideal
the
-adic completion of
is an image of a formal power series ring, specifically, the image of the surjection
::
:The kernel is the ideal
Completions can also be used to analyze the local structure of
singularities of a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
. For example, the affine schemes associated to
and the nodal cubic
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
have similar looking singularities at the origin when viewing their graphs (both look like a plus sign). Notice that in the second case, any Zariski neighborhood of the origin is still an irreducible curve. If we use completions, then we are looking at a "small enough" neighborhood where the node has two components. Taking the localizations of these rings along the ideal
and completing gives
and
respectively, where
is the formal square root of
in
More explicitly, the power series:
:
Since both rings are given by the intersection of two ideals generated by a homogeneous degree 1 polynomial, we can see algebraically that the singularities "look" the same. This is because such a scheme is the union of two non-equal linear subspaces of the affine plane.
Properties
- The completion of a Noetherian ring with respect to some ideal is a Noetherian ring.
- The completion of a Noetherian local ring with respect to the unique maximal ideal is a Noetherian local ring.
- The completion is a functorial operation: a continuous map ''f'': ''R'' → ''S'' of topological rings gives rise to a map of their completions,
Moreover, if ''M'' and ''N'' are two modules over the same topological ring ''R'' and ''f'': ''M'' → ''N'' is a continuous module map then ''f'' uniquely extends to the map of the completions:
:
where are modules over
- The completion of a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
''R'' is a flat module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact se ...
over ''R''.
- The completion of a finitely generated module ''M'' over a Noetherian ring ''R'' can be obtained by ''extension of scalars'':
:
Together with the previous property, this implies that the functor of completion on finitely generated ''R''-modules is exact: it preserves
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
s. In particular, taking quotients of rings commutes with completion, meaning that for any quotient ''R''-algebra , there is an isomorphism
:
Cohen structure theorem In mathematics, the Cohen structure theorem, introduced by , describes the structure of complete Noetherian local rings.
Some consequences of Cohen's structure theorem include three conjectures of Krull:
*Any complete regular equicharacteristic ...
(equicharacteristic case). Let ''R'' be a complete local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
Noetherian commutative ring with maximal ideal and residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
''K''. If ''R'' contains a field, then
:
for some ''n'' and some ideal ''I'' (Eisenbud, Theorem 7.7).
See also
* Formal scheme
*Profinite integer In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)
:\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p
where
:\varprojlim \mathbb/n\mathbb
indicates the profinite completion of \math ...
*Locally compact field In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space.. These kinds of fields were originally introduced in p-adic analysis since the fields \mathbb_p are locally compact topological spac ...
*Zariski ring In commutative algebra, a Zariski ring is a commutative Noetherian topological ring ''A'' whose topology is defined by an ideal \mathfrak a contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced by under the ...
*Linear topology
In algebra, a linear topology on a left A-module M is a topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretc ...
* Quasi-unmixed ring
Citations
References
*
* David Eisenbud, ''Commutative algebra. With a view toward algebraic geometry''. Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...
, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ;
* Fujiwara, K.; Gabber, O.; Kato, F.:
On Hausdorff completions of commutative rings in rigid geometry
” ''Journal of Algebra'', 322 (2011), 293–321.
{{refend
Commutative algebra
Topological algebra