# Origin of the term “localization” for the localization of a ring

I'm curious if the term localization in ring theory comes from algebraic geometry or not. The connection between localization and "looking locally about a point" seems like it should be the source for the notion of localization. It seems plausible, but it seems like we would have had to wait until Zariski defined the Zariski topology for the connection to become apparent. That seems hard to believe given the amount of work done in commutative algebra before 20th century, especially given the importance of localization in commutative algebra.

Then this raises the question: Where and when was the term 'localization' first used to describe the adjunction of inverses, and does it originate from algebraic geometry or from somewhere else? Was the notion of localization used regularly with a different name before it was given this name?

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See the Historical Notes of Bourbaki's Commutative Algebra. I believe they say the general notion of localization (not just domains) was isolated by Uzkov in 1940 or so. – KConrad Apr 19 '10 at 22:46
General category construction is due to Gabriel and Gabriel-Zisman – Shizhuo Zhang Apr 19 '10 at 22:48
Why is this community wiki? Surely there ought to be an answer to this question. – José Figueroa-O'Farrill Apr 19 '10 at 23:22
Well, there's the obvious stupid answer: on an affine scheme, restriction to distinguished open sets corresponds to localization of the ring. It seems rather clear that localization is a good name for this, especially since you can look at smaller and smaller open sets around a point. Everything else (e.g. stalks being localization) can be understood as the same idea taken to extremes (limits). – Ilya Grigoriev Apr 20 '10 at 1:48
Dear Harry, As an aside: the role of localization as a technical tool in commutative algebra is due to Bourbaki, I think. If you look in Zariski--Samuel, say, it does not play the same role. One certainly shouldn't be looking back before the 20th century (when very little abstract algebra existed), but rather in the middle (loosely speaking) of the 20th century. (Based on the dates in my answer below, and the date provided by Keith Conrad, I would say that the answer lies in the literature between the 1930s and the 1960s.) – Emerton Apr 20 '10 at 17:19

I'm looking at the paper "On the theory of local rings" by Chevalley (Annals of Math. 44 (1943)). In this paper he explains how to localize at a multiplicative set $S$ of non-zero divisors, and calls this the ring of quotients of the set $S$.

There is no question that Chevalley was motivated by algebraic geometry.

The paper "Generalized semi-local rings", by Zariski (Summa Brasiliensis Math. 1 (1946)) attributes the theory of local rings to Krull (in a paper called "Dimensionstheorie in Stellenringen", Crelle 179 (1938), which I don't have a copy of at hand) and Chevalley (in the above mentioned paper), so it seems that the Chevalley reference above is a reasonable guide to the situation.

Of course none of these references quite address the origin of the term localization at $S$, but (based on my prior preconceptions, and bolstered by having looked at these two papers) I am fairly confident that it was indeed motivated by algebraic geometry.

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Very interesting. I would have thought that localization was a much earlier invention (especially because it is in principle much easier to define than the stalk of a structure sheaf of a locally ringed space). – Harry Gindi Apr 20 '10 at 18:06

I don't remember the history too well, but the answers above perfectly fit one of my favorite quotes from V.I. Arnold on this very question that illustrates the gulf between (1) axiomatic training and (2) hands-on approach.

1. there's the obvious stupid answer: on an affine scheme, restriction to distinguished open sets corresponds to localization of the ring. It seems rather clear that localization is a good name for this, especially since you can look at smaller and smaller open sets around a point. (Ilya Grigoriev)

2. If $M$ is a manifold and $x\in M$ , then the ring of smooth germs in $x$ is canonically isomorphic to the localization […] Geometric localization is expressed algebraically in introducing inverses for functions for which it makes sense, thus you can call it localization. (Martin Brandenburg)

Студенты высшей нормальной школы в Париже спросили меня: "Почему вы называете кольцо формальных степенных рядов локальным? Разве оно удовлетворяет аксиомам локального кольца?" Для неспециалистов поясню, что заданный вопрос аналогичен вопросу "Почему вы называете окружность коническим сечением?" Это были лучшие студенты-математики Франции. По-видимому, какой-то преступный алгебраист обучил их аксиомам колец (и даже локальных колец), не приводя ни одного примера (и, в частности, не объяснив происхождение термина "локальное").

(В.И. Арнольд, Топологические проблемы теории распространения волн, УМН, т.51, вып.1 (307), 1996, с.5)

Here is my rather literal translation:

Students from École Normale Supérieure, Paris asked me: "Why are you referring to the formal power ring as local? Does it really satisfy axioms of a local ring?" Let me remark for the non-experts that their question is analogous to the question: "Why do you call the cirlce a conic section?" Those were the best mathematics students in France. Apparently, some criminal algebraist taught them ring axioms (and even local ring axioms) without giving a single example (and, in particular, without explaining the origin of the term "local").

(V.I. Arnold, Topological problems in the theory of wave propagation, Russian Math Surveys, 51:1, 1996)

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Well, Arnold famously dislikes axiomatics not grounded in concrete examples, especially coming from abstract algebra. The paragraph preceding the one which appears above is funnier (although not related to local rings): during a written exam, in Paris, a student told Arnold that he forgot his calculator and could Arnold tell him if 4/7 was greater or less than 1. (Convergence of an integral depended on this.) Arnold generally compared this knowledge-without-examples in higher math to Feynman's discussion of "Brazilian" physics in his Surely You're Joking book. – KConrad May 25 '10 at 17:08
Victor, as you know, though related, the notion of localization from the question is not the same as the notion of a local ring. Typical localization used for Zariski open subsets look geometrically and intuitively radically different from the localization at a point leading to a local ring; the latter are about the formal/infinitesimal neighborhood -- in my opinion much more abstract notion than the Zariski open sets. Your preference to localization at point seems based on liking the personality of Arnol'd who was systematically rude to any misgiving of French students and mathematicians. – Zoran Skoda May 15 '11 at 16:55

If $M$ is a manifold and $x \in M$, then the ring of smooth germs in $x$ is canonical isomorphic to the localization $C^{\infty}(M)_{\mathfrak{p}}$, where $\mathfrak{p} = \{f \in C^{\infty}(M) : f(x) = 0\}$. I believe this was known long before the Zariski topology. And yet you get the same message: Geometric localization is expressed algebraically in introducing inverses for functions for which it makes sense, thus you can call it localization.

I'm also interested in a historical source, but I don't think that the terminology emerged from algebraic geometry. It's at least one instance for motivating this terminology, among othes such as differential geometry and also functional analysis.

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Martin- This is a history question, and yet your answer doesn't cite any sources. You seem to just be saying it's equally possible that the term came from differential geometry. – Ben Webster Apr 20 '10 at 16:58
Right . – Martin Brandenburg Apr 20 '10 at 17:10
@Martin: localizing at a multiplicative set containing zero divisors (as in your example!) was not introduced until Uzkov around 1940's; even Chevalley in 1943 didn't use such generality, according to Emerton's reference. Your belief that such an equality was known "long before Zariski" seems to be a wild guess, moreover contrary to the historical evidence on the algebra side. Is there any evidence for it? Why would pre-Zariski geometers or function theorists even want such a result? The ring of germs is a quotient of $C^{\infty}(M)$, so why would they bother with abstract localization? – Boyarsky Jun 29 '10 at 11:11