Localization (commutative algebra)

In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units (invertible elements) in R*. Further one wants R* to be the 'best possible' or 'most general' way to do this—in the usual fashion this should be expressed by a universal property. The localization of R by S is usually denoted by S⁻¹R; however other notations are used in some important special cases. If S is the set of the non zero elements of an integral domain, then the localization is the field of fractions and thus usually denoted Frac(R). If S is the complement of a prime ideal I the localization is denoted by R_I, and R_f is used to denote the localization by the powers of an element f.^[1] The two latter cases are fundamental in algebraic geometry and scheme theory. In particular the definition of an affine scheme is based on the properties of these two kinds of localizations.

Terminology

The term localization originates 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 R* contains only information about the behavior of V near p. For more detail, see Ring of germs.

In number theory and algebraic topology, one refers to the behavior of a ring at a number n or away from n. "Away from n" means "in the ring localized by the set of the powers of n" (which is a Z[1/n]-algebra). If n is a prime number, "at n" means "in the ring localized by the set of the integers that are not a multiple of n".

Construction and properties for commutative rings

The set S is assumed to be a submonoid of the multiplicative monoid of R, i.e. 1 is in S and for s and t in S we also have st in S. A subset of R with this property is called a multiplicatively closed set or more briefly, a multiplicative set. This requirement on S is natural and necessary to have since its elements will be turned into units of the localization, and units must be closed under multiplication.

It is standard practice to assume that S is multiplicatively closed. If S is not multiplicatively closed, it suffices to replace it by its multiplicative closure, consisting of the set of the products of elements of S (including the empty product 1). This does not change the result of the localization. The fact that we talk of "a localization with respect to the powers of an element" instead of "a localization with respect to an element" is an example of this. Therefore, we shall suppose S to be multiplicatively closed in what follows.

Construction

For integral domains

In the case R is an integral domain there is an easy construction of the localization. Since the only ring in which 0 is a unit is the trivial ring {0}, the localization R* is {0} if 0 is in S. Otherwise, the field of fractions K of R can be used: we take R* to be the subset of K consisting of the elements of the form r/s with r in R and s in of S; as we have supposed S multiplicatively closed, R* is a subring. The standard embedding of R into R* is injective in this case, although it may be non-injective in a more general setting. For example, the dyadic fractions are the localization of the ring of integers with respect to the powers of two. In this case, R* is the dyadic fractions, R is the integers, the denominators are powers of 2, and the natural map from R to R* is injective. The result would be exactly the same if we had taken S={2}.

For general commutative rings

For general commutative rings, we don't have a field of fractions. Nevertheless, a localization can be constructed consisting of "fractions" with denominators coming from S; in contrast with the integral domain case, one can safely 'cancel' from numerator and denominator only elements of S.

This construction proceeds as follows: on R × S define an equivalence relation ~ by setting (r₁,s₁) ~ (r₂,s₂) if there exists t in S such that

t(r₁s₂ − r₂s₁) = 0.

(The presence of t is crucial to the transitivity of ~)

We think of the equivalence class of (r,s) as the "fraction" r/s and, using this intuition, the set of equivalence classes R* can be turned into a ring with operations that look identical to those of elementary algebra: a/s + b/t = (at + bs)/st and (a/s)(b/t) = ab/st. The map j : R → R* that maps r to the equivalence class of (r,1) is then a ring homomorphism. In general, this is not injective; if a and b are two elements of R such that there exists s in S with s(a − b) = 0, then their images under j are equal.

Universal property

The above-mentioned universal property is the following: the ring homomorphism j : R → R* maps every element of S to a unit in R*, and if f : R → T is some other ring homomorphism which maps every element of S to a unit in T, then there exists a unique ring homomorphism g : R* → T such that f = g ∘ j.

Examples

Given a commutative ring R, we can consider the multiplicative set S of non-zero-divisors (i.e. elements a of R such that multiplication by a is an injection from R into itself.) The ring S⁻¹R is called the total quotient ring of R. S is the largest multiplicative set such that the canonical mapping from R to S⁻¹R is injective. When R is an integral domain, this is the fraction field of R.
The ring Z/nZ where n is composite is not an integral domain. When n is a prime power it is a finite local ring, and its elements are either units or nilpotent. This implies it can be localized only to a zero ring. But when n can be factorised as ab with a and b coprime and greater than 1, then Z/nZ is by the Chinese remainder theorem isomorphic to Z/aZ × Z/bZ. If we take S to consist only of (1,0) and 1 = (1,1), then the corresponding localization is Z/aZ.
Let R = Z, and p a prime number. If S = Z - pZ, then R* is the localization of the integers at p. See Lang's "Algebraic Number Theory," especially pages 3–4 and the bottom of page 7.
As a generalization of the previous example, let R be a commutative ring and let p be a prime ideal of R. Then R - p is a multiplicative system and the corresponding localization is denoted R_p. The unique maximal ideal is then pR_p.
For the commutative ring $\mathbb {C} [x,y]$ its localization for the maximal ideal $(x,y)$ is the ring of rational fractions $\mathbb {C} [x,y]_{(x,y)}=\{f/g:f,g\in \mathbb {C} [x,y]{\text{ and }}g(0,0)\neq 0\}$ .
Let R be a commutative ring and f an element of R. we can consider the multiplicative system {fⁿ : n = 0,1,...}. Then the localization intuitively is just the ring obtained by inverting powers of f. If f is nilpotent, the localization is the zero ring.

Properties

Some properties of the localization R* = S⁻¹R:

S⁻¹R = {0} if and only if S contains 0.
The ring homomorphism R → S⁻¹R is injective if and only if S does not contain any zero divisors.
There is a bijection between the set of prime ideals of S⁻¹R and the set of prime ideals of R which do not intersect S. This bijection is induced by the given homomorphism R → S⁻¹R.
In particular: after localization at a prime ideal P, one obtains a local ring, or in other words, a ring with one maximal ideal, namely the ideal generated by the extension of P.

Category theoretic description

Another way to describe the localization of a ring R at a subset S is via category theory. If R is a ring and S is a subset, consider all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. These algebras are the objects of a category, with R-algebra homomorphisms as morphisms. Then, the localization of R at S is the initial object of this category. (This is a more abstract way of expressing the universal property above.)

Applications

Two classes of localizations occur commonly in commutative algebra and algebraic geometry and are used to construct the rings of functions on open subsets in Zariski topology of the spectrum of a ring, Spec(R).

The set S consists of all powers of a given element r. The localization corresponds to restriction to the Zariski open subset U_r ⊂ Spec(R) where the function r is non-zero (the sets of this form are called principal Zariski open sets). For example, if R = K[X] is the polynomial ring and r = X then the localization produces the ring of Laurent polynomials K[X, X⁻¹]. In this case, localization corresponds to the embedding U ⊂ A¹, where A¹ is the affine line and U is its Zariski open subset which is the complement of 0.
The set S is the complement of a given prime ideal P in R. The primality of P implies that S is a multiplicatively closed set. In this case, one also speaks of the "localization at P". Localization corresponds to restriction to arbitrary small open neighborhoods of the irreducible Zariski closed subset V(P) defined by the prime ideal P in Spec(R).

Non-commutative case

Localizing non-commutative rings is more difficult. While the localization exists for every set S of prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved is the Ore condition.

One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D⁻¹ for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.

References

^ Eisenbud, Harris, The geometry of schemes

Cohn, P. M. (1989). "§ 9.3". Algebra. Vol. Vol. 2 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xvi+428. ISBN 0-471-92234-X. MR 1006872. {{cite book}}: |volume= has extra text (help)
Cohn, P. M. (1991). "§ 9.1". Algebra. Vol. Vol. 3 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xii+474. ISBN 0-471-92840-2. MR 1098018. {{cite book}}: |volume= has extra text (help)
Stenström, Bo (1971). Rings and modules of quotients. Lecture Notes in Mathematics, Vol. 237. Berlin: Springer-Verlag. pp. vii+136. ISBN 978-3-540-05690-4. MR 0325663.
Serge Lang, "Algebraic Number Theory," Springer, 2000. pages 3–4.

External links

Localization from MathWorld.

[1] Eisenbud, Harris, The geometry of schemes

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