Localization in a commutative algebra
A transition from a commutative ring to the ring of fractions (cf. Fractions, ring of)
, where
is a subset of
. The ring
can be defined as the solution of the problem of a universal mapping from
into a ring under which all elements of
become invertible. However, there are explicit constructions for
:
1) as the set of fractions of the form , where
and
is a product of elements of
(two fractions
and
are regarded as equivalent if and only if there is an
that is a product of elements of
and is such that
; fractions are added and multiplied by the usual rules);
2) as the quotient ring of the ring of polynomials ,
, with respect to the ideal generated by the polynomials
,
;
3) as the inductive limit of an inductive system of -modules
, where
runs through a naturally-ordered free commutative monoid
. All the
are isomorphic to
, and the homomorphisms
with
coincide with multiplication by
.
The ring is canonically mapped into
and converts the latter into an
-algebra. This mapping
is injective if and only if
does not contain any divisor of zero in
. On the other hand, if
contains a nilpotent element, then
.
Without loss of generality the set can be assumed to be closed with respect to products (such a set is known as multiplicative, or as a multiplicative system). In this case the ring
is also denoted by
or
. The most important examples of multiplicative systems are the following:
a) the set of all powers of an element of
;
b) the set , that is, the complement of a prime ideal
. The corresponding ring of fractions is local and is denoted by
;
c) the set of all non-divisors of zero in
.
The ring is called the complete ring of fractions of
. If
is integral, then
is a field of fractions.
The operation of localization carries over with no difficulty to arbitrary -modules
if one sets
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The transition from to
is an exact functor. In other words, the
-module
is flat. Localization commutes with direct sums and inductive limits.
From the geometrical point of view localization means transition to an open subset. More precisely, for the spectrum
is canonically identified with the open (in the Zariski topology) subset
consisting of the prime ideals
not containing
. Moreover, this operation makes it possible to associate with each
-module
a quasi-coherent sheaf
on the affine scheme
for which
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Localization can be regarded as an operation that makes it possible to invert morphisms of multiplication by an in the category of
-modules. In this approach the operation of localization admits a wide generalization to arbitrary categories (see Localization in categories).
References
[1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
Localization in a commutative algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Localization_in_a_commutative_algebra&oldid=18943