localization of a commutative ring
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The localization of a commutative ring at a set of its elements is a new ring in which the elements of become invertible (units) and which is universal with this property.
When interpreting a ring under Isbell duality as the ring of functions on some space (its spectrum), then localizing it at corresponds to restricting to the complement of the subspace on which the elements in vanish.
See also commutative localization and localization of a ring (noncommutative).
For commutative rings
Let be a commutative ring. Let be a multiplicative subset of the underlying set.
The following gives the universal property of the localization.
The localization is a homomorphism to another commutative ring such that
for all elements the image is invertible (is a unit);
for every other homomorphism with this property, there is a unique homomorphism such that we have a commuting diagram
The following gives an explicit description of the localization
For a commutative ring and an element, the localization of at is
(e.g. Sullivan 70, first pages)
(…) By the lifting property of etale morphisms for -rings, see here. (…)
A classical set of lecture notes is
- Dennis Sullivan, Localization, Periodicity and Galois Symmetry (The 1970 MIT notes) edited by Andrew Ranicki, K-Monographs in Mathematics, Dordrecht: Springer (pdf)
Other accounts of the basics include
Revised on August 18, 2014 21:44:12
by Urs Schreiber