Origin of the term "localization" for the localization of a ring - MathOverflow

Origin of the term "localization" for the localization of a ring

2010-04-19T22:41:57Z

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?

Answer by Martin Brandenburg for Origin of the term "localization" for the localization of a ring

2010-04-20T09:04:38Z

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.

Answer by Emerton for Origin of the term "localization" for the localization of a ring

2010-04-20T16:53:10Z

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.

Answer by Victor Protsak for Origin of the term "localization" for the localization of a ring

2010-05-25T08:32:58Z

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.

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)
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)