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 m s , {\displaystyle {\frac {m}{s}},} 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 Q {\displaystyle \mathbb {Q} } of rational numbers from the ring Z {\displaystyle \mathbb {Z} } 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 S − 1 R {\displaystyle S^{-1}R} 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 (cf. the example given at local ring).
Localization of a ring The localization of a commutative ring R by a multiplicatively closed set S is a new ring S − 1 R {\displaystyle S^{-1}R} 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.
Multiplicative set 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. The localization by a set U that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of U. However, the same localization is obtained by using the multiplicatively closed set S of all products of elements of U. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets. For example, the localization by a single element s introduces fractions of the form a s , {\displaystyle {\tfrac {a}{s}},} but also products of such fractions, such as a b s 2 . {\displaystyle {\tfrac {ab}{s^{2}}}.} So, the denominators will belong to the multiplicative set { 1 , s , s 2 , s 3 , … } {\displaystyle \{1,s,s^{2},s^{3},\ldots \}} of the powers of s. Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element". The localization of a ring R by a multiplicative set S is generally denoted S − 1 R , {\displaystyle S^{-1}R,} but other notations are commonly used in some special cases: if S = { 1 , t , t 2 , … } {\displaystyle S=\{1,t,t^{2},\ldots \}} consists of the powers of a single element, S − 1 R {\displaystyle S^{-1}R} is often denoted R t ; {\displaystyle R_{t};} if S = R ∖ p {\displaystyle S=R\setminus {\mathfrak {p}}} is the complement of a prime ideal p {\displaystyle {\mathfrak {p}}} , then S − 1 R {\displaystyle S^{-1}R} is denoted R p . {\displaystyle R_{\mathfrak {p}}.}
In the remainder of this article, only localizations by a multiplicative set are considered.
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