Definition
In this article, let R be a commutative ring and M an R-module.
Let S a multiplicatively closed subset of R, i.e. for any s and t ∈ S, the product st is also in S. Then the localization of M with respect to S, denoted S−1M, is defined to be the following module: as a set, it consists of equivalence classes of pairs (m, s), where m ∈ M and s ∈ S. Two such pairs (m, s) and (n, t) are considered equivalent if there is a third element u of S such that
- u(sn-tm) = 0
It is common to denote these equivalence classes
- .
To make this set a R-module, define
and
(a ∈ R). It is straightforward to check that the definition is well-defined, i.e. yields the same result for different choices of representatives of fractions. One interesting characterization of the equivalence relation is that it is the smallest relation (considered as a set) such that cancellation laws hold for elements in S. That is, it is the smallest relation such that rs/us = r/u for all s in S.
One case is particularly important: if S equals the complement of a prime ideal p ⊂ R (which is multiplicatively closed by definition of prime ideals) then the localization is denoted Mp instead of (R\p)−1M. The support of the module M is the set of prime ideals p such that Mp ≠ 0. Viewing M as a function from the spectrum of R to R-modules, mapping
this corresponds to the support of a function. Localization of a module at primes also reflects the "local properties" of the module. In particular, there are many cases where the more general situation can be reduced to a statement about localized modules. The reduction is because a R-module M is trivial if and only if all its localizations at primes or maximal ideals are trivial.
Read more about this topic: Localization Of A Module
Famous quotes containing the word definition:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)