Completion (Commutative Algebra)

The completion or completion of a ring or a module is a technique in commutative algebra , in which a ring or a module is completed with respect to a certain metric, which is usually induced by an ideal. The term is geometrically related to the localization of a ring: Both ring extensions examine the vicinity of a point in the spectrum of a ring , but the completion reflects the local appearance even more.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

Completion of a ring in relation to an ideal

Be a ring and an ideal. ${\displaystyle A}$ ${\displaystyle I}$

In the ring

${\displaystyle A^{\mathbb {N} }=\prod _{n\in \mathbb {N} }A}$

becomes a consequence

${\displaystyle (a_{i})_{(i\in \mathbb {N} )}=(a_{0},a_{1},\dots )}$

Zero sequence called when all one is, so that: ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k\in \mathbb {N} }$

${\displaystyle \forall i>k:a_{i}\in I^{n}}$

${\displaystyle \mathrm {NF} }$ be the ideal of all null sequences.

One episode

${\displaystyle (a_{i})_{(i\in \mathbb {N} )}}$

is Cauchy called when all one is, so that: ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k\in \mathbb {N} }$

${\displaystyle \forall i,j>k:(a_{i}-a_{j})\in I^{n}}$

${\displaystyle \mathrm {CF} }$ be the subring of all Cauchy episodes

The ring

${\displaystyle {\hat {A}}_{I}:=\mathrm {CF} /\mathrm {NF} }$

is defined as the completion of relative designated. ${\displaystyle A}$ ${\displaystyle I}$

For is ${\displaystyle a\in A}$

${\displaystyle (a,a,a\dots )}$

a Cauchy series.

The image

${\displaystyle f\colon A\to {\hat {A}}}$

${\displaystyle f\colon a\mapsto (a,a,\dots )}$

is injective if and only if:

${\displaystyle \bigcap _{i=0}^{\infty }I^{i}=\{0\}}$

The ring is called complete ( complete ) (with respect to ) if is an isomorphism. ${\displaystyle I}$ ${\displaystyle f}$

Examples

Formal power series

If the polynomial ring is over a field and is the ideal , Cauchy sequences of polynomials correspond to infinite polynomials ${\displaystyle A}$ ${\displaystyle K[X_{1},\dots ,X_{n}]}$ ${\displaystyle I}$ ${\displaystyle (X_{1},\dots ,X_{n})}$

${\displaystyle \sum _{i=0}^{\infty }\alpha _{i}x^{i}}$

The completion of is isomorphic to the ring of formal power series ${\displaystyle K[X_{1},\dots ,X_{n}]}$ ${\displaystyle K[[X_{1},\dots ,X_{n}]]}$

P-adic numbers

The p-adic numbers are described as the completion of with respect to the -adic metric : are and rational numbers with ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle p}$ ${\displaystyle d_{p}}$ ${\displaystyle q}$ ${\displaystyle r}$

${\displaystyle q-r=\pm \ p^{i}\cdot {\dfrac {s}{t}}}$

with and and does not share , so is ${\displaystyle s,t\in \mathbb {N} }$ ${\displaystyle i\in \mathbb {Z} }$ ${\displaystyle p}$ ${\displaystyle s,t}$

${\displaystyle d_{p}(q-r)=p^{-i}}$

A sequence of integers is a Cauchy sequence with respect to the -adic metric if and only if it is a Cauchy sequence with respect to the ideal . One therefore obtains an embedding: ${\displaystyle p}$ ${\displaystyle (p)}$

${\displaystyle f\colon {\widehat {\mathbb {Z} }}_{p}\hookrightarrow \mathbb {Q} _{p}}$.

Here, the left side denotes the completion of regarding . This embedding even provides an isomorphism to the ring of the whole p-adic numbers . Because of Hensel's lemma, there are many non-rational algebraic numbers, e.g. B. the -th roots of unity. ${\displaystyle \mathbb {Z} }$ ${\displaystyle (p)}$ ${\displaystyle {\widehat {\mathbb {Z} _{p}}}\cong \mathbb {Z} _{p}}$ ${\displaystyle {\widehat {\mathbb {Z} }}_{p}}$ ${\displaystyle (p-1)}$

Geometric example

The Newtonian knot in the real affine plane

Let be the plane algebraic curve in two-dimensional affine space given by the equation ${\displaystyle X}$

${\displaystyle y^{2}=x^{2}(x+1)}$

is defined. The curve intersects at the zero point. It is called the Newtonian node and looks around the zero point (clearly) locally like the curve that is given by the equation: ${\displaystyle Y}$

${\displaystyle 0=xy}$ is defined.

This geometric fact corresponds to the isomorphism:

${\displaystyle {\hat {A}}\cong {\hat {B}}}$

With

${\displaystyle A:=K[[x,y]]/(y^{2}-x^{2}-x^{3})_{({\bar {x}},{\bar {y}})}}$

and

${\displaystyle B:=K[[x,y]]/(xy)_{({\bar {x}},{\bar {y}})}}$

The local rings of the points are not isomorphic, but their completions with respect to their maximum ideals are.

The ring on the left-hand side of the “isomorphism equation” is also an example that the completion of an integrity range does not have to be an integrity range.

From an analytical point of view, the Newtonian knot as a subset of the complex plane as a whole is irreducible, but locally divides into two branches around zero. Because for is the root of holomorphic, so you can write: ${\displaystyle x<1}$ ${\displaystyle (x+1)}$

${\displaystyle y^{2}-x^{2}(x+1)=(y+x{\sqrt {x+1}})(y-x{\sqrt {x+1}})=f_{1}\cdot f_{2}}$

with two holomorphic functions and . ${\displaystyle f_{1}}$ ${\displaystyle f_{2}}$

Algebraic-geometric interpretation

The importance of completion for algebraic geometry is that in the completed ring one can study the local appearance of the variety. If two points and two irreducible varieties have isomorphic local rings, then the varieties and are already birationally equivalent . The local ring already contains almost all information about the variety, while the completion of the local ring comes closer to the intuition about local behavior. ${\displaystyle P\in X}$ ${\displaystyle Q\in Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

The following sentence applies:

Be a noetherian local ring with the maximum ideal and its completion. Then: ${\displaystyle A}$ ${\displaystyle m}$ ${\displaystyle {\hat {A}}}$

• ${\displaystyle \mathrm {dim} (A)=\mathrm {dim} ({\hat {A}})}$
• ${\displaystyle A}$is regular if and only if it is. ${\displaystyle {\hat {A}}}$

Cohen's structural theorem makes a statement about the completion of local rings of varieties:

If a regular local ring is complete with respect to its maximum ideal and contains a body, then: ${\displaystyle A}$

${\displaystyle A\cong k[[X_{1},\dots ,X_{n}]]}$

where is the remainder field of . ${\displaystyle k}$ ${\displaystyle A}$

Regular points on algebraic varieties of the same dimension have isomorphic completions, just as points on manifolds of the same dimensions have homeomorphic neighborhoods.

Functorial properties

Are and rings and as well as ideals and ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle I\subset A}$ ${\displaystyle J\subset B}$

${\displaystyle f\colon A\to B}$

a ring homomorphism with:

${\displaystyle f(I)\subset J}$

(Such a ring homomorphism is called continuous) then there is a homomorphism

${\displaystyle {\hat {f}}\colon {\hat {A}}\to {\hat {B}}}$

" " Is therefore a functor with continuous mappings as morphisms ${\displaystyle \wedge }$

Construction alternatives and generalizations

Generalizations on modules through filters

Filtration of a module is a consequence ${\displaystyle M}$

${\displaystyle (M_{i})_{i\in \mathbb {N} }}$

so that

${\displaystyle M=M_{0}\supset M_{1}\supset ...\supset M_{i}\supset }$

They now play the role of in the definition of null sequence and Cauchy sequence . The definitions can be carried over literally. It is ${\displaystyle M_{i}}$ ${\displaystyle I^{n}}$

${\displaystyle {\hat {M}}=CF/NF}$

and is called complete (with regard to the filtration) if the figure ${\displaystyle M}$

${\displaystyle f\colon M\to {\hat {M}}}$

is an isomorphism.

Rings as (pseudo) metric spaces

The completion of a ring with respect to an ideal can be understood as a special case of the completion of a metric space if a suitable metric is defined on the ring.

If a ring is and an ideal, a distance can be defined for this ring through the ideal by: ${\displaystyle A}$ ${\displaystyle I}$

${\displaystyle \mathrm {d} (x,y)=\mathrm {inf} \{2^{-i}|x-y\in I^{i}\}\ ({\text{ mit }}I^{0}:=A)}$

This is a pseudometric because:

${\displaystyle \mathrm {d} (x,y)\geq 0}$

${\displaystyle \mathrm {d} (x,x)=0}$

${\displaystyle \mathrm {d} (x,y)=\mathrm {d} (y,x)}$

${\displaystyle \mathrm {d} (x,y)\leq \mathrm {d} (x,z)+\mathrm {d} (z,y)}$

If:

${\displaystyle \bigcap _{i=0}^{\infty }I^{i}=\{0\},}$

so the distance function is a metric; i.e., it also applies:

${\displaystyle \mathrm {d} (x,y)=0\Rightarrow x=y}$

With regard to this (pseudo) metric, the above-mentioned terms Cauchy sequence, zero sequence and completion agree with those of the metric spaces.

Completion as an inverse limes

An inverse system of rings (or modules) is (here) a sequence of rings (or modules) and homomorphisms

${\displaystyle (A_{i},f_{i})_{(i\in \mathbb {N} )}}$

so that

${\displaystyle f_{n}\colon A_{n}\to A_{n-1}}$

So:

${\displaystyle \dots {\xrightarrow {\ f_{3}\ }}A_{2}{\xrightarrow {\ f_{2}\ }}A_{1}{\xrightarrow {\ f_{1}\ }}A_{0}}$

The inverse limit of this inverse system is:

${\displaystyle \lim _{\longleftarrow }(A_{n},f_{n})_{n\in \mathbb {N} }:={\biggl \{}(x_{n})_{n\in \mathbb {N} }\in \prod _{n\in \mathbb {N} }A_{n}{\biggl |}x_{n}\in A_{n},f_{n}(x_{n})=x_{n-1}{\biggl \}}}$

Is now an ideal and ${\displaystyle I\subset A}$

${\displaystyle A_{i}=A/I^{i}}$

${\displaystyle A_{0}=0}$

${\displaystyle f_{i+1}\colon A_{i+1}\to A_{i}}$

${\displaystyle f_{i+1}\colon {\bar {x}}\mapsto {\bar {x}}}$ (Whereby different, i.e. the corresponding remainder classes are meant.)

then the following isomorphism applies:

${\displaystyle {\hat {A}}_{I}\cong \lim _{\longleftarrow }(A_{n},f_{n})_{n\in \mathbb {N} }}$

literature

• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Atiyah, Macdonald: Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
• Robin Hartshorne : Algebraic Geometry , Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9