nLab
Ore localization

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Definition

If SRS\subset R is a left Ore set in a monoid (or a ring) RR, then we call the pair (j,S 1R)(j,S^{-1}R) where j:RS 1Rj:R\to S^{-1}R is a morphism of monoids (rings) the left Ore localization of RR with respect to SS if it is the universal object in the category C=C(R,S)C = C(R,S) whose objects are the pairs (f,Y)(f,Y) where f:RYf : R \rightarrow Y is a morphism of rings from RR into a ring YY such that the image f(S)f(S) of SS consists of units (=multiplicatively invertible elements), and the morphisms α:(f,Y)(f,Y)\alpha : (f,Y) \rightarrow (f',Y') are maps of rings α:YY\alpha : Y \rightarrow Y' such that αf=f\alpha \circ f = f'.

The definition of CC makes sense even if SRS\subset R is not left Ore; the universal object in CC may then exist when SS is not left Ore, for example this is the case when SS is right Ore, while not left Ore. In fact, the universal object is a left Ore localization (i.e. SS is left Ore) iff it lies in the full subcategory C lC^l of CC whose objects (f,Y)(f,Y) satisfy two additional conditions:

(i) f(S) 1f(R)={(f(s)) 1f(r)|sS,rR}f(S)^{-1}f(R) = \{(f(s))^{-1}f(r)\,|\, s \in S, r\in R\} is a subring in YY,

(ii) kerf=I Sker\,f = I_S.

Hence (j,S 1R)(j,S^{-1}R) is universal in C lC^l, and that characterizes it, but the universality in CC, although not characteristic, appears to be more useful in practice.

For every left Ore set SRS\subset R in a monoid or ring RR, the left Ore localization exists and it can be defined as follows. As a set, S 1R:=S×R/S^{-1}R := S\times R/\sim, where \sim is the following relation of equivalence:

(1)(s,r)(s,r)(s˜Sr˜R)(s˜s=r˜sands˜r=r˜r). (s,r) \sim (s',r') \,\,\Leftrightarrow\,\, (\exists \tilde s\in S \,\,\exists \tilde{r}\in R) \,\, (\tilde{s}s' = \tilde{r}s\,\,and\,\,\tilde{s}r' = \tilde{r}r).

A class of equivalence of (s,r)(s,r) is denoted s 1rs^{-1}r and called a left fraction. The multiplication is defined by s 1 1r 1s 2 1r 2=(s˜s 1) 1(r˜r 2)s_1^{-1}r_1\cdot s_2^{-1}r_2 = (\tilde{s}s_1)^{-1} (\tilde{r}r_2) where r˜R,s˜S\tilde{r} \in R, \tilde{s} \in S satisfy r˜s 2=s˜r 1\tilde{r}s_2 = \tilde{s}r_1 (one should think of this, though it is not yet formally justified at this point, as s 1r˜=r 1s 2 1s^{-1}\tilde{r} = r_1 s_2^{-1}, what enables to put inverses one next to another and then the multiplication rule is obvious). If the monoid RR is a ring then we can extend the addition to S 1RS^{-1}R too. Suppose we are given two fractions with representatives (s 1,r 1)(s_1,r_1) and (s 2,r 2)(s_2,r_2). Then by the left Ore condition we find s˜S\tilde{s} \in S, r˜R\tilde{r}\in R such that s˜s 1=r˜s 2\tilde{s} s_1 = \tilde{r} s_2. The sum is then defined

(2)s 1 1r 1+s 2 1r 2:=(s˜s 1) 1(s˜r 1+r˜r 2) s_1^{-1} r_1 + s_2^{-1} r_2 \,:=\, (\tilde{s}s_1)^{-1} (\tilde{s}r_1 + \tilde{r}r_2)

It is a long and at points tricky to work out all the details of this definition. One has to show that \sim is indeed relation of equivalence, that the operations are well defined, and that S 1RS^{-1}R is indeed a ring. Even the commutativity of the addition needs work (there is an alternative definition of addition in which s˜\tilde{s} above is not required to be in SS but the product s˜r 1\tilde{s}r_1 is in SS; this approach is manifestly commutative but it has some other drawbacks). At the end, one shows that the map j=j S:RS 1Rj = j_S : R \rightarrow S^{-1}R given by i(r)=1 1ri(r) = 1^{-1}r is a homomorphism of rings, which is 1-1 iff the 2-sided ideal I S={nR|sS,sn=0}I_S = \{ n \in R \,|\,\exists s \in S,\, sn = 0\} is zero.

Basic property of Ore localization is flatness: S 1RS^{-1}R is flat RR-bimodule. The left Ore localization j:RS 1Rj : R\to S^{-1}R induced a flat localization functor for the category of left RR-modules or the category of right RR-modules. The localization functor is the extension of scalars along l S:RS 1Rl_S: R\to S^{-1}R, that is (in the case of left RR-modules) MS 1R RMM\mapsto S^{-1}R\otimes_R M is the flat localization functor Q S *: RMod S 1RModQ^*_S:{}_R Mod\to {}_{S^{-1}R}Mod and the restriction of scalars is its right adjoint Q S*Q_{S*} which is fully faithful and it has its own right adjoint (the localization functor is affine).

References

  • K. R. Goodearl, Robert B. Warfield, An introduction to noncommutative Noetherian rings, London Math. Soc. Student Texts 16 (1st ed,), 1989, xviii+303 pp.; or 61 (2nd ed.), 2004, xxiv+344 pp.

  • Zoran Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (pdf), ed. A. Ranicki; pp. 220–313, math.QA/0403276.

Revised on July 19, 2014 01:43:59 by Urs Schreiber (82.113.106.79)