Localization (commutative algebra)

In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field of rational numbers from the ring of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term localization originated in algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety "locally" near a point p, then one considers the set S of all functions that are not zero at p and localizes R with respect to S. The resulting ring contains information about the behavior of V near p, and excludes information that is not "local", such as the zeros of functions that are outside V (c.f. the example given at local ring). The localization of a commutative ring R by a multiplicatively closed set S is a new ring whose elements are fractions with numerators in R and denominators in S. If the ring is an integral domain the construction generalizes and follows closely that of the field of fractions, and, in particular, that of the rational numbers as the field of fractions of the integers. For rings that have zero divisors, the construction is similar but requires more care. Localization is commonly done with respect to a multiplicatively closed set S (also called a multiplicative set or a multiplicative system) of elements of a ring R, that is a subset of R that is closed under multiplication, and contains 1. The requirement that S must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to S.

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