If $S\subset R$ is a left Ore set in a monoid (or a ring) $R$, then we call the pair $(j,S^{-1}R)$ where $j:R\to S^{-1}R$ is a morphism of monoids (rings) the left Ore localization of $R$ with respect to $S$ if it is the universal object in the category $C = C(R,S)$ whose objects are the pairs $(f,Y)$ where $f : R \rightarrow Y$ is a morphism of rings from $R$ into a ring $Y$ such that the image $f(S)$ of $S$ consists of units (=multiplicatively invertible elements), and the morphisms $\alpha : (f,Y) \rightarrow (f',Y')$ are maps of rings $\alpha : Y \rightarrow Y'$ such that $\alpha \circ f = f'$.
The definition of $C$ makes sense even if $S\subset R$ is not left Ore; the universal object in $C$ may then exist when $S$ is not left Ore, for example this is the case when $S$ is right Ore, while not left Ore. In fact, the universal object is a left Ore localization (i.e. $S$ is left Ore) iff it lies in the full subcategory $C^l$ of $C$ whose objects $(f,Y)$ satisfy two additional conditions:
(i) $f(S)^{-1}f(R) = \{(f(s))^{-1}f(r)\,|\, s \in S, r\in R\}$ is a subring in $Y$,
(ii) $ker\,f = I_S$.
Hence $(j,S^{-1}R)$ is universal in $C^l$, and that characterizes it, but the universality in $C$, although not characteristic, appears to be more useful in practice.
For every left Ore set $S\subset R$ in a monoid or ring $R$, the left Ore localization exists and it can be defined as follows. As a set, $S^{-1}R := S\times R/\sim$, where $\sim$ is the following relation of equivalence:
A class of equivalence of $(s,r)$ is denoted $s^{-1}r$ and called a left fraction. The multiplication is defined by $s_1^{-1}r_1\cdot s_2^{-1}r_2 = (\tilde{s}s_1)^{-1} (\tilde{r}r_2)$ where $\tilde{r} \in R, \tilde{s} \in S$ satisfy $\tilde{r}s_2 = \tilde{s}r_1$ (one should think of this, though it is not yet formally justified at this point, as $s^{-1}\tilde{r} = r_1 s_2^{-1}$, what enables to put inverses one next to another and then the multiplication rule is obvious). If the monoid $R$ is a ring then we can extend the addition to $S^{-1}R$ too. Suppose we are given two fractions with representatives $(s_1,r_1)$ and $(s_2,r_2)$. Then by the left Ore condition we find $\tilde{s} \in S$, $\tilde{r}\in R$ such that $\tilde{s} s_1 = \tilde{r} s_2$. The sum is then defined
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 $S^{-1}R$ is indeed a ring. Even the commutativity of the addition needs work (there is an alternative definition of addition in which $\tilde{s}$ above is not required to be in $S$ but the product $\tilde{s}r_1$ is in $S$; this approach is manifestly commutative but it has some other drawbacks). At the end, one shows that the map $j = j_S : R \rightarrow S^{-1}R$ given by $i(r) = 1^{-1}r$ is a homomorphism of rings, which is 1-1 iff the 2-sided ideal $I_S = \{ n \in R \,|\,\exists s \in S,\, sn = 0\}$ is zero.
One defines a localization functor which is the extension of scalars $Q^*_S = S^{-1}R\otimes_R - : R-mod\to S^{-1}R-mod$, $M\mapsto S^{-1}R\otimes_R M$. The localization functor is exact (“flat”), has a fully faithful right adjoint, namely the restriction of scalars $Q_{S*}$ and the latter has its own right adjoint (the localization functor is affine). In particular, it realizes $S^{-1}R-mod$ as a reflective subcategory of $R-mod$ and the composition endofunctor $Q_{S*} Q^*_S$ is underlying the corresponding idempotent monad in $R-mod$. The component of the unit of its adjunction $\eta_R:R\to S^{-1} R$ equals the canonical localization map $j$ and $\eta_M = j\otimes_R\id_M$.
Given any multiplicative set $S\subset R$, the set of all left ideals $I\subset R$ such that $\forall r\in R$ $\{z\in R| z r\in I\}\cap S\neq 0$ is a Gabriel filter $\mathcal{F}_S$. If $S$ is left Ore it is sufficient to ask that $I\cap S\neq\emptyset$. The Gabriel localization functor corresponding to this filter is isomorphic to $Q^*_S$ if $S$ is left Ore.
Basic property of Ore localization is flatness: $S^{-1}R$ is a flat $R$-bimodule.
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.
Last revised on August 15, 2024 at 15:37:36. See the history of this page for a list of all contributions to it.