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created localization of a ring
Once i have time (this week probably no) then I will rewrite the entry. Namely the definition there applies only to certain localizations, it does not apply, for example, for very important Cohn universal localization of a ring (over there one in fact inverts certain matrices over a ring). In ring theory there are many kinds of localization concepts. I will be happy to expand on this, but I will have to change as well.
Now I want to make changes to the entry.
I see now that the entry is now called localization of a commutative ring and localization of a ring is a redirect. I wrote a more extensive entry in early days of my $n$Lab involvement called commutative localization what is a bit more general than localization of a commutative ring; namely it is a localization at a central multiplicative set. I also wrote some time ago Ore localization and a stub for Cohn localization aka Cohn localization (both can be understood as localization of noncommutative unital rings).
First of all I will separate localization of a ring from localization of a commutative ring.
I have created a new stub localization of a ring different from localization of a commutative ring. Note also field of fractions (which refers to commutative localization).
Entries with word localization in the title so far: Beilinson-Bernstein localization, Bousfield localization, Bousfield localization of model categories, Bousfield localization of triangulated categories, Gabriel localization, Ore localization, affine localization, [[algebraic microlocalization, cohomology localization, commutative localization, compatible localization,equivariant localization, equivariant localization and elimination of nodes, homology localization, homotopy localization, iterated localization, localization, localization of a commutative ring, localization of a model category, localization of a module, localization of a ring, localization of a simplicial model category, localization of an (infinity,1)-category, localization of an abelian category, localization of an enriched category, microlocalization, noncommutative localization, reflective localization, simplicial localization, simplicial localization of a homotopical category, strict localization, topological localization, universal localization
I have formatted slightly at localization of a ring and added the half-sentence on what characterizes the localization map.
Thanks, though, again, as I explained earlier, this (what you added) is true for commutative localization and Ore localization but it is not appropriate for Cohn localization (where matrices are inverted) nor for Gabriel localization of a ring itself at a filter $\mathcal{F}_S$ obtained from a multiplicative set $S$ (not all kinds of noncommutative ring localizations, are having this ring theoretic universal property). I added a sentence modifying a bit this statement. I think more precise statement should not be in that entry, but rather at entries dedicated to special kinds like Cohn localization, Ore localization, Martindale localization, Gabriel localization…
Okay. Though calling Cohn localization a localization of the ring instead of of its category of modules is a bit of an abuse of language. But all right.
I have edited Cohn localization a little. Is it wise to have the entry titled “universal localization”? That seems a very ambiguous title for a very specific entry.
Urs, Cohn localization is yet a stub. Cohn localization is most often in ring theory viewed and used only as a localization of ring and not as its associated torsion theory/localization functor !
Namely, take a multiplicative set of matrices $\Sigma$ (let us not dwell on the definition of it, it is a set of matrices of varying sizes satisfying some closure properties) over a ring $R$.
We say that a map of rings $f: R\to S$ is $\Sigma$-inverting if all entries of $f(A)$ for each $A\in \Sigma$ are invertible in $S$. The Cohn localization of a ring $R$, is a homomorphism of rings $R\to \Sigma^{-1} R$ which is initial in the category of all $\Sigma$-inverting maps (which is the subcategory of coslice category $R/Ring$).
Universal localization is an entirely standard term. Notice that every subset $S\subset R$ can be viewed as generating some multiplicative set of matrices of various sizes (the need for latter is to fake the Ore property by sort of Gauss elimination procedure). The Ore localization is a special case or Cohn localization.
Universal localization is an entirely standard term.
In ring theory. You see, there are subfields of mathematics where even a bare “abc” is an entirely standard term, as we have just seen. But when we are talking to a potential audience that consists of more than a small group of specialists, we should try to use more self-contained terms.
Given all the entries that we have on the notion of “localization”, as you have listed above, given all the many many fields that these apply to, and given that about every notion of localization is universal in some sense, it seems unreasonable to declare that the words “universal localization” should refer exclusively to a very particular case of a localization in a very particular context only.
How about “universal localization of a ring”, at least?
In ring theory.
And in torsion theory of abelian categories, in skewfields, in universal algebra community, and in the setup of triangulated categories and in topology, and also in descent theory. See volume edited by Ranicki to get some feeling of how spread it is.
I agree that it is rather pompous to call it universal (it has clean universal property what some ring theoretic localizations do not, as I emphasise today!) as there are many other “universal” localizations in different setup. But it is named so by Cohn, and then accepted by algebraic topologists like Vogel, Ranicki, Neeman, by algebraists like Bokut and then it spread widely. I like to call it Cohn localization what is also somewhat used (in algebra but not so much in topology where universal is predominant for that notion).
So you are definitely opposed to renaming the entry to “universal localization of a ring”?
Read above. It is sometimes viewed as kind of a torsion theory and ring is not that clearly seen in that case. Sometimes one looks at localization of a ring and sometimes at associated functor. in both cases people call it universal localization or Cohn localization or Cohn universal localization. And it has some generalizations.
Another interesting thing is that Cohn localization may be viewed (by a result of Vogel) as $H^0$ of certain Bousfield localization of a triangulated category.
Though the term is not the happiest, we do not intend to ever use the technical term universal localization for some other localization than Cohn’s, do we ? I mean many localizations have universal properties but we do not say that the localization itself is universal, so we do not need to free the term anyway… Besides, Cohn may have been motivated by the fact that he can localize for every $S$ and even $\Sigma$; other kind of taking quotients in ring theory needed special assumptions on inverting set, or did not have a universal property. This one is universal in three different senses – first in the sense for every $S$ and $\Sigma$ it exists, and besides it has a clean universal property at the level of rings; and it also has another universal property at the level of full categories of modules (because it is a torsion theory).
I won’t further argue. But I record that I am opposed to using “universal X” in an entry title for something that is far from being universal among $X$s.
It is universally applicable (for all subsets of $R$, unlike any other noncommutative localization); in that sense it is universal among all noncommutative localizations. (true, Gabriel’s localization is also for every $S$, via filter $\mathcal{F}_S$ but it does not really invert in such formal sense). Funny enough Cohn called it “inversive localization” in what I think is the first article he wrote on the topic; this is even worse terminology, comparing to most modern localization concepts which are, of course, also inversive.
(Unfortunately I do not have access to this article (neither from IRB nor from IHES). Anybody can send me the pdf ?)
However, there are also partial version where one inverts just matrices from the left (or right), which he called semi-invertive in this paper. They are of some use in algebra.
According to this Wikipedia article, the localization of a noncommutative ring cannot always be formed as described. The relation used to define the equivalence classes is not necessarily transitive in the non-commutative case. The provided reference to the Stacks project is in a section contained in the chapter on commutative algebra. That is why I made this change.
The Wikipedia article I linked mentions the Ore condition as a sufficient condition for this, which I see was mentioned earlier in this discussion, and has its own nlab page here: Ore localization. Perhaps a more nuanced discussion about when the given definition does apply to non-commutative rings is warranted.
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