Localization (commutative algebra): Difference between revisions

Revision as of 04:50, 12 January 2006

Revision as of 04:54, 12 January 2006

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:''t''(''r''₁''s''₂ − ''r''₂''s''₁) = 0.

We think of the [[equivalence class]] of (''r'',''s'') as the "fraction" ^''r''⁄_''s'', and, using this intuition, the set of equivalence classes ''R*'' can be turned into a ring; the map ''j'' : ''R'' → ''R*'' which maps ''r'' to the equivalence class of (''r'',1) is then a [[ring homomorphism]].

The above mentioned universal property is the following: the ring homomorphism ''j'' : ''R'' → ''R*'' maps every element of ''S'' to a unit in ''R*'', and if ''f'' : ''R'' → ''T'' is some other ring homomorphism which maps every element of ''S'' to a unit in ''T'', then there exists a unique ring homomorphism ''g'' : ''R*'' → ''T'' such that ''f'' = ''g'' o ''j''.

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Localizing non-commutative rings is more difficult; the localization does not exist for every set ''S'' of prospective units. One condition which ensures that the localization exists is the [[Ore condition]].

One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D^-1 for a differentiation operator D. This is done in many contexts in methods for [[differential equation]]s. There is now a large mathematical theory about it, named [[microlocalization]], connecting with numerous other branches. The ''micro-'' tag is to do with connections with [[Fourier theory]], in particular.

== See also ==