commutative localization

Given a unital ring (or only a monoid) RR and a central multiplicative subset SZ(R)RS\subset Z(R)\subset R (i.e. set containing 1R1\in R and with every two elements containing their product, and such that all its elements are central in RR) the ring of fractions (monoid of fractions, respectively) S 1RS^{-1} R is sometimes said to be the commutative localization of RR at SS; the same name is also given to the canonical map RS 1RR\to S^{-1} R of rings (monoids resp.). The ring of fractions is defined as the set of equivalence classes (s,r)S×R(s,r)\in S\times R where (s,r)(t,r)(s,r)\sim (t,r') iff uS\exists u\in S, usr=utru s r' = u t r (if RR is an integral domain one can skip mentioning uu in this condition); the equivalence classes are called fractions and denoted s 1rs^{-1}r; by centrality of SS it is easy to guess the multiplication rule s 1rt 1r=(ts) 1(rr)s^{-1}r t^{-1} r' = (t s)^{-1} (r r') and for the addition one first takes the representatives with the same denominator and then adds the numerators. E.g. the formula s 1r+t 1r=(ts) 1(tr+sr)s^{-1}r + t^{-1}r' = (t s)^{-1} (t r + s r') will do, and we indeed get a ring S 1RS^{-1} R with unit 1 111^{-1} 1 together with the canonical homomorphism of rings RS 1RR\to S^{-1} R given by r1 1rr\mapsto 1^{-1} r.

Localization of commutative rings at multiplicative subsets is the standard example, but the centrality of SS is enough for the whole theory to pass through.

Commutative localization can be extended to left modules.

Module of fractions S 1MS^{-1} M is the left S 1RS^{-1} R-module S 1MS^{-1} M equipped with the natural map of RR-modules MS 1MM\to S^{-1}M and defined as follows:

The underlying set of S 1MS^{-1} M consists of equivalence classes s 1ms^{-1} m of pairs (s,m)S×M(s,m)\in S\times M where (s,m)(t,n)(s,m)\cong (t,n) iff there exist uSu\in S such that utm=usnu t m = u s n, the multiplication by scalar is defined by (s 1r)(t 1m):=(ts) 1(rm)(s^{-1} r)(t^{-1} m):= (t s)^{-1} (r m) and the addition is s 1m+t 1n:=(st) 1(tm+sn)s^{-1} m + t^{-1} n := (s t)^{-1}(t m + s n). The correspondence Q S:MS 1MQ_S : M\mapsto S^{-1} M extends to a functor RMod S 1RMod{}_R Mod\to {}_{S^{-1} R}Mod. The forgetful functor U: S 1RMod RModU: {}_{S^{-1} R}Mod \to {}_R Mod is fully faithful functor and there is a natural transformation of functors IdUQ SId\to U Q_S whose components are the RR-module maps MS 1MM\to S^{-1}M given by m1 1mm\mapsto 1^{-1}m.

It can be then shown that this elementary approach is equivalent to the definition via the extension of scalars formula S 1M=S 1R RMS^{-1}M = S^{-1} R\otimes_R M.

The basic result is that the commutative localization S 1RS^{-1} R is a flat left module over RR, the property which holds for more general Ore localization.

Commutative localization in which also RR is commutative is a basic procedure used in defining algebraic scheme as a locally ringed space. Another special case of this procedure is forming the quotient field of a commutative integral domain.

Revised on November 25, 2013 02:28:25 by Urs Schreiber (