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Completion of a ring

In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, 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 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 with Cauchy sequences, and agrees with it in the case when R has a metric given by a non-Archimedean absolute value.

General construction edit

Suppose that E is an abelian group with a descending filtration

 

of subgroups. One then defines the completion (with respect to the filtration) as the inverse limit:

 

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, a filtered module, or a filtered vector space, 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 and noncommutative rings. As may be expected, when the intersection of the   equals zero, this produces a complete[clarification needed] topological ring.

Krull topology edit

In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal   is especially important, for example the distinguished maximal ideal of a valuation ring. The basis of open neighbourhoods of 0 in R is given by the powers In, which are nested and form a descending filtration on R:

 

(Open neighborhoods of any rR are given by cosets r + In.) The completion is the inverse limit of the factor rings,

 

pronounced "R I hat". The kernel of the canonical map π from the ring to its completion is the intersection of the powers of I. Thus π is injective if and only if this intersection reduces to the zero element of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a local ring.

There is a related topology on R-modules, also called Krull or I-adic topology. A basis of open neighborhoods of a module 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 over  .

Examples edit

  • The ring of p-adic integers   is obtained by completing the ring   of integers at the ideal (p).
  • Let R = K[x1,...,xn] be the polynomial ring in n variables over a field K and   be the maximal ideal generated by the variables. Then the completion   is the ring K[[x1,...,xn]] 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[1]
 
The kernel is the ideal  

Completions can also be used to analyze the local structure of singularities of a scheme. For example, the affine schemes associated to   and the nodal cubic plane curve   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 edit

  • The completion of a Noetherian ring with respect to some ideal is a Noetherian ring.[2]
  • The completion of a Noetherian local ring with respect to the unique maximal ideal is a Noetherian local ring.[3]
  • The completion is a functorial operation: a continuous map fR → 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 fM → 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 R is a flat module over R.[4]
  • 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 sequences. In particular, taking quotients of rings commutes with completion, meaning that for any quotient R-algebra  , there is an isomorphism

 
  • Cohen structure theorem (equicharacteristic case). Let R be a complete local Noetherian commutative ring with maximal ideal   and residue field K. If R contains a field, then
 
for some n and some ideal I (Eisenbud, Theorem 7.7).

See also edit

Citations edit

  1. ^ "Stacks Project — Tag 0316". stacks.math.columbia.edu. Retrieved 2017-01-14.
  2. ^ Atiyah & Macdonald 1969, Theorem 10.26.
  3. ^ Atiyah & Macdonald 1969, Proposition 10.16. and Theorem 10.26.
  4. ^ Atiyah & Macdonald 1969, Proposition 10.14.

References edit

completion, ring, other, uses, completeness, abstract, algebra, completion, several, related, functors, rings, modules, that, result, complete, topological, rings, modules, completion, similar, localization, together, they, among, most, basic, tools, analysing. For other uses see Completeness In abstract algebra a completion is any of several related functors on rings and modules that result in complete topological rings and modules Completion is similar to localization 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 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 with Cauchy sequences and agrees with it in the case when R has a metric given by a non Archimedean absolute value Contents 1 General construction 2 Krull topology 3 Examples 4 Properties 5 See also 6 Citations 7 ReferencesGeneral construction editSuppose that E is an abelian group with a descending filtration E F 0 E F 1 E F 2 E displaystyle E F 0 E supset F 1 E supset F 2 E supset cdots nbsp of subgroups One then defines the completion with respect to the filtration as the inverse limit E lim E F n E displaystyle widehat E varprojlim E F n E nbsp 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 a filtered module or a filtered vector space 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 and noncommutative rings As may be expected when the intersection of the F i E displaystyle F i E nbsp equals zero this produces a complete clarification needed topological ring Krull topology editMain article Adic topology In commutative algebra the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull after Wolfgang Krull or I adic topology on R The case of a maximal ideal I m displaystyle I mathfrak m nbsp is especially important for example the distinguished maximal ideal of a valuation ring The basis of open neighbourhoods of 0 in R is given by the powers In which are nested and form a descending filtration on R F 0 R R I I 2 F n R I n displaystyle F 0 R R supset I supset I 2 supset cdots quad F n R I n nbsp Open neighborhoods of any r R are given by cosets r In The completion is the inverse limit of the factor rings R I lim R I n displaystyle widehat R I varprojlim R I n nbsp pronounced R I hat The kernel of the canonical map p from the ring to its completion is the intersection of the powers of I Thus p is injective if and only if this intersection reduces to the zero element of the ring by the Krull intersection theorem this is the case for any commutative Noetherian ring which is an integral domain or a local ring There is a related topology on R modules also called Krull or I adic topology A basis of open neighborhoods of a module M is given by the sets of the form x I n M for x M displaystyle x I n M quad text for x in M nbsp The completion of an R module M is the inverse limit of the quotients M I lim M I n M displaystyle widehat M I varprojlim M I n M nbsp This procedure converts any module over R into a complete topological module over R I displaystyle widehat R I nbsp Examples editThe ring of p adic integers Z p displaystyle mathbb Z p nbsp is obtained by completing the ring Z displaystyle mathbb Z nbsp of integers at the ideal p Let R K x1 xn be the polynomial ring in n variables over a field K and m x 1 x n displaystyle mathfrak m x 1 ldots x n nbsp be the maximal ideal generated by the variables Then the completion R m displaystyle widehat R mathfrak m nbsp is the ring K x1 xn of formal power series in n variables over K Given a noetherian ring R displaystyle R nbsp and an ideal I f 1 f n displaystyle I f 1 ldots f n nbsp the I displaystyle I nbsp adic completion of R displaystyle R nbsp is an image of a formal power series ring specifically the image of the surjection 1 R x 1 x n R I x i f i displaystyle begin cases R x 1 ldots x n to widehat R I x i mapsto f i end cases nbsp dd The kernel is the ideal x 1 f 1 x n f n displaystyle x 1 f 1 ldots x n f n nbsp Completions can also be used to analyze the local structure of singularities of a scheme For example the affine schemes associated to C x y x y displaystyle mathbb C x y xy nbsp and the nodal cubic plane curve C x y y 2 x 2 1 x displaystyle mathbb C x y y 2 x 2 1 x nbsp 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 x y displaystyle x y nbsp and completing gives C x y x y displaystyle mathbb C x y xy nbsp and C x y y u y u displaystyle mathbb C x y y u y u nbsp respectively where u displaystyle u nbsp is the formal square root of x 2 1 x displaystyle x 2 1 x nbsp in C x y displaystyle mathbb C x y nbsp More explicitly the power series u x 1 x n 0 1 n 2 n 1 2 n n 2 4 n x n 1 displaystyle u x sqrt 1 x sum n 0 infty frac 1 n 2n 1 2n n 2 4 n x n 1 nbsp 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 editThe completion of a Noetherian ring with respect to some ideal is a Noetherian ring 2 The completion of a Noetherian local ring with respect to the unique maximal ideal is a Noetherian local ring 3 The completion is a functorial operation a continuous map f R S of topological rings gives rise to a map of their completions f R S displaystyle widehat f widehat R to widehat S nbsp 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 f M N displaystyle widehat f widehat M to widehat N nbsp where M N displaystyle widehat M widehat N nbsp are modules over R displaystyle widehat R nbsp The completion of a Noetherian ring R is a flat module over R 4 The completion of a finitely generated module M over a Noetherian ring R can be obtained by extension of scalars M M R R displaystyle widehat M M otimes R widehat R nbsp Together with the previous property this implies that the functor of completion on finitely generated R modules is exact it preserves short exact sequences In particular taking quotients of rings commutes with completion meaning that for any quotient R algebra R I displaystyle R I nbsp there is an isomorphism R I R I displaystyle widehat R I cong widehat R widehat I nbsp Cohen structure theorem equicharacteristic case Let R be a complete local Noetherian commutative ring with maximal ideal m displaystyle mathfrak m nbsp and residue field K If R contains a field thenR K x 1 x n I displaystyle R simeq K x 1 ldots x n I nbsp for some n and some ideal I Eisenbud Theorem 7 7 See also editFormal scheme Profinite integer Locally compact field Zariski ring Linear topology Quasi unmixed ringCitations edit Stacks Project Tag 0316 stacks math columbia edu Retrieved 2017 01 14 Atiyah amp Macdonald 1969 Theorem 10 26 Atiyah amp Macdonald 1969 Proposition 10 16 and Theorem 10 26 Atiyah amp Macdonald 1969 Proposition 10 14 References editAtiyah Michael Francis Macdonald I G 1969 Introduction to Commutative Algebra Westview Press ISBN 978 0 201 40751 8 David Eisenbud Commutative algebra With a view toward algebraic geometry Graduate Texts in Mathematics 150 Springer Verlag New York 1995 xvi 785 pp ISBN 0 387 94268 8 ISBN 0 387 94269 6 MR1322960 Fujiwara K Gabber O Kato F On Hausdorff completions of commutative rings in rigid geometry Journal of Algebra 322 2011 293 321 Retrieved from https en wikipedia org w index php title Completion of a ring amp oldid 1183927382, wikipedia, wiki, book, books, library,

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