Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorjcmckeown
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexThe convention, when describing ring extensions, everywhere I've seen a convention, is that * for $S$ a set of primes, "localize at $S$" means "invert what is not divisible by $S$"; so for $p$ prime, localizing "at $p$" means considering only $p$-torsion. * adjoining inverses $[S^{-1}]$ is pronounced "localized away from $S$". Inverting a prime $p$ is localizing away from $p$, which means *ignoring* $p$-torsion. I have adjusted four instances of former "at" on three pages that would be, algebraicwise, "away from" (and so they now appear). Evidently, this conflicts with more-categorial uses of "localized"; "inverting weak equivalences" is called [[localization]], by obvious analogy, and is written as "localizing **at** weak equivalences". This is confusing! It's also weird: since a ring is a one-object $Ab$-enriched category with morphisms "multiply-by", the localization-of-the-category $R$ "at $p$" (or its $Ab$-enriched version, if saying that is necessary) *really* means the localization-of-the-ring $R$ "away from $p$". You all can sort out that contravariance as/if you like, but don't break the old algebra papers!

   The convention, when describing ring extensions, everywhere I’ve seen a convention, is that

   • for $S$ a set of primes, “localize at $S$” means “invert what is not divisible by $S$”; so for $p$ prime, localizing “at $p$” means considering only $p$-torsion.
   • adjoining inverses $[S^{-1}]$ is pronounced “localized away from $S$”. Inverting a prime $p$ is localizing away from $p$, which means ignoring $p$-torsion.

   I have adjusted four instances of former “at” on three pages that would be, algebraicwise, “away from” (and so they now appear).

   Evidently, this conflicts with more-categorial uses of “localized”; “inverting weak equivalences” is called localization, by obvious analogy, and is written as “localizing at weak equivalences”. This is confusing! It’s also weird: since a ring is a one-object $Ab$-enriched category with morphisms “multiply-by”, the localization-of-the-category $R$ “at $p$” (or its $Ab$-enriched version, if saying that is necessary) really means the localization-of-the-ring $R$ “away from $p$”.

   You all can sort out that contravariance as/if you like, but don’t break the old algebra papers!

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, probably what you fixed were my edits. I had been thinking about adding such a kind of discussion somewhere to the lab, but then didn't. But now I pasted your paragraph into [this entry here](http://ncatlab.org/nlab/show/localization+of+a+ring#Terminology)! :-)

   Thanks, probably what you fixed were my edits. I had been thinking about adding such a kind of discussion somewhere to the lab, but then didn’t.

   But now I pasted your paragraph into this entry here! :-)

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeNov 23rd 2013
   • (edited Nov 23rd 2013)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexAs a specialist in noncommutative localization, I strongly disagree with extending the conventions from localization mod p to the general case. This mod p thing is very special and it is done mainly in computational rather than geometric work. In noncommutatve ring theory, we localize at Ore set, and the geometric meaning is dual: localizing away from singularities of inverses, that is away from the point where inverses are not defined. You are inverting elements of Ore sets what makes sense AT places where the inverses make sense; it is tautology. On the other hand, the terminology localizing at a prime, is generalizing to localizing at prime ideal, in nc case at completely prime ideal, what is Ore condition for the complement. But this is a very special localization where you go away from the whole space except for one point. In geometry you localize to get an OPEN set, so you are away from something of lower codimension.

   As a specialist in noncommutative localization, I strongly disagree with extending the conventions from localization mod p to the general case. This mod p thing is very special and it is done mainly in computational rather than geometric work. In noncommutatve ring theory, we localize at Ore set, and the geometric meaning is dual: localizing away from singularities of inverses, that is away from the point where inverses are not defined. You are inverting elements of Ore sets what makes sense AT places where the inverses make sense; it is tautology.

   On the other hand, the terminology localizing at a prime, is generalizing to localizing at prime ideal, in nc case at completely prime ideal, what is Ore condition for the complement. But this is a very special localization where you go away from the whole space except for one point. In geometry you localize to get an OPEN set, so you are away from something of lower codimension.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeNov 23rd 2013
   • (edited Nov 23rd 2013)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexUrs, I want to roll back a part of what you pasted there: it gives impression that even in the cases of algebraic localizaton which have nothing to do with prime ideals one uses this prime terminology. On the contrary, in the noncommutative localization, except for primes, we use the categorical terminology. I put quite a lot of time to a large circle of entries in nLab concerning noncommutative localization (e.g. topologizing subcategories, Serre subcategories, Ore localization etc.) and this is now bringing confusion.

   Urs, I want to roll back a part of what you pasted there: it gives impression that even in the cases of algebraic localizaton which have nothing to do with prime ideals one uses this prime terminology. On the contrary, in the noncommutative localization, except for primes, we use the categorical terminology. I put quite a lot of time to a large circle of entries in nLab concerning noncommutative localization (e.g. topologizing subcategories, Serre subcategories, Ore localization etc.) and this is now bringing confusion.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeNov 23rd 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI think Zoran and jcmckeown are actually saying the same thing: when we say "localize at S" and "away from S", S is not a set of elements of the ring but a set of points on the associated geometric object. So, there is no conflict if we're careful about the type of S.

   I think Zoran and jcmckeown are actually saying the same thing: when we say “localize at S” and “away from S”, S is not a set of elements of the ring but a set of points on the associated geometric object. So, there is no conflict if we’re careful about the type of S.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeNov 25th 2013
   • PermaLink
   Author: zskoda
   Format: MarkdownItexSurely, there is no question of the geometric meaning, Marc, I agree that we agree there. The question is of the traditional terminology points of view in various parts of localization theory.

   Surely, there is no question of the geometric meaning, Marc, I agree that we agree there. The question is of the traditional terminology points of view in various parts of localization theory.

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeDec 21st 2013
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexSo Zoran, how would you [write this](http://ncatlab.org/nlab/show/localization+of+a+ring#Terminology)?

   So Zoran, how would you write this?