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The convention, when describing ring extensions, everywhere I’ve seen a convention, is that
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!
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! :-)
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.
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.
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.
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.
So Zoran, how would you write this?
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