Localization (algebra)

In commutative algebra and algebraic geometry, the 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 one so that it consists of fractions

{\frac {m}{s}},

such that the denominator s belongs to a given subset S of R. The basic example is the construction of the ring Q of rational numbers from the ring Z of rational integers.

The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, 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 which 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. Cf. the example given at local ring.

An important related process is completion: one often localizes a ring/module, then completes.

In this article, a ring is commutative with unity.

Construction

Localization of 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.

Let S be a multiplicatively closed subset of a ring R, i.e. for any s and t ∈ S, the product st is also in S, and $1\in S$ . Then the localization of R with respect to S, denoted S⁻¹R, is defined to be the following ring: as a set, it consists of equivalence classes of pairs (m, s), where m ∈ R 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

(The presence of u is crucial to the transitivity of ~. If $0\in S,$ then every pair is equivalent to (0, 1), and the localization is the zero ring; for this reason, this case is excluded by some authors.) It is common to denote these equivalence classes

{\frac {m}{s}}

Thus, S consists of "denominators".

To make this set a ring, define

{\frac {m}{s}}+{\frac {n}{t}}:={\frac {tm+sn}{st}}

and

{\frac {m}{s}}{\frac {n}{t}}:={\frac {mn}{st}}

It is straightforward to check that the definition is well-defined, i.e. independent of choices of representatives of fractions. One then checks that the two operations are in fact addition and multiplication (associativity, etc) and that they are compatible (that is, distribution law). This step is also straightforward. The zero element is $0/1$ and the unity is $1/1$ ; they are usually simply denoted by 0 and 1.

Finally, there is a canonical map $j:R\to S^{{-1}}R,m\mapsto m/1$ . (In general, it is not injective; if two elements of R differ by a nonzero zero-divisor with an annihilator in S, they have the same image by very definition.) The above mentioned universal property is the following: j : R → R* maps every element of S to a unit in R* (since (1/s)(s/1) = 1), 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

If R has no nonzero zero-divisors (i.e., R is an integral domain), then the equivalence (m, s) ~ (n, t) reduces to

sn = tm

which is precisely the condition we get when we formally clear out the denominators in ${\frac {m}{s}}={\frac {n}{t}}$ . This motivates the definition above. In fact, the localization recovers the construction of the field of fractions as follows. Since the zero ideal is prime, its complement S is multiplicatively closed. The localization $S^{{-1}}R$ then consists of $r/s,r\in R,s\in R\setminus \{0\}$ . That is, $S^{{-1}}R$ is precisely the field of fractions K of R. Since there is no nonzero zero-divisor, the canonical map $m\to m/1$ is an inclusion and one can view R as a subring of K. Indeed, any localization of an integral domain is a subring of the field of fractions (cf. overring).

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 R_p. If S consists of all powers of a nonzero nilpotent f, then $S^{-1}R$ is denoted by either $R_{f}$ or $R[f^{{-1}}].$

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 the set of all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. The elements of this set form 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.

Localization of a module

The construction above applies to a module $M$ over a ring $R$ except that instead of multiplication we define the scalar multiplication by

a\cdot {\frac {m}{s}}:={\frac {am}{s}}

Then $S^{{-1}}M$ is a $R$ -module consisting of $m/s$ with the operations defined above. As above, there is a canonical module homomorphism

φ: M → S⁻¹M

mapping

φ(m) = m / 1.

The same notations for the localization of a ring are used for modules: $M_{\mathfrak {p}}$ denote the localization of M at a prime ideal ${\mathfrak {p}}$ and $M_{f}$ the localization of a non-nilpotent element f.

By the very definitions, the localization of the module is tightly linked to the one of the ring via the tensor product

S⁻¹M = M ⊗_RS⁻¹R,

This way of thinking about localising is often referred to as extension of scalars.

As a tensor product, the localization satisfies the usual universal property.

Examples and applications

Given a commutative ring R, we can consider the multiplicative set S of non-zerodivisors (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 none other than 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.
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 p.
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.

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 the complement U in Spec(R) of the irreducible Zariski closed subset V(P) defined by the prime ideal P.

Properties

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

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.

The localization of a module $M\to S^{{-1}}M$ is a functor from the category of R-modules to the category of $S^{-1}R$ -modules. From the definition, one can see that it is exact, or in other words (reading this in the tensor product) that S⁻¹R is a flat module over R. This is actually foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of an open set in Spec(R) (see spectrum of a ring) is a flat morphism.

The localization functor (usually) preserves Hom and tensor products in the following sense: the natural map

S^{{-1}}(M\otimes _{R}N)\to S^{{-1}}M\otimes _{{S^{{-1}}R}}S^{{-1}}N

is an isomorphism and if $M$ is finitely presented, the natural map

S^{{-1}}\operatorname {Hom}_{R}(M,N)\to \operatorname {Hom}_{{S^{{-1}}R}}(S^{{-1}}M,S^{{-1}}N)

is an isomorphism.

If a module M is a finitely generated over R, we have: $S^{{-1}}M=0$ if and only if $tM=0$ for some $t\in S$ if and only if $S$ intersects the annihilator of M.^[1]

Let R be an integral domain with the field of fractions K. Then its localization $R_{\mathfrak {p}}$ at a prime ideal ${\mathfrak {p}}$ can be viewed as a subring of K. Moreover,

R=\cap _{{\mathfrak {p}}}R_{{\mathfrak {p}}}=\cap _{{\mathfrak {m}}}R_{{\mathfrak {m}}}

where the first intersection is over all prime ideals and the second over the maximal ideals.^[2]

Let ${\sqrt {I}}$ denote the radical of an ideal I in R. Then

{\sqrt {I}}\cdot S^{{-1}}R={\sqrt {I\cdot S^{{-1}}R}}

In particular, R is reduced if and only if its total ring of fractions is reduced.^[3]

Stability under localization

Many properties of a ring are stable under localization. For example, the localization of a noetherian ring (resp. principal ideal domain) is noetherian (resp. principal ideal domain). The localization of an integrally closed domain is an integrally closed domain. In many cases, the converse also holds. (See below)

Local property

Let M be a R-module. We could think of two kinds of what it means some property P holds for M at a prime ideal ${\mathfrak {p}}$ . One means that P holds for $M_{\mathfrak {p}}$ ; the other means that P holds for a neighborhood of ${\mathfrak {p}}$ . The first interpretation is more common.^[4] But for many properties the first and second interpretations coincide. Explicitly, the second means the following conditions are equivalent.

(i) P holds for M.
(ii) P holds for $M_{\mathfrak {p}}$ for all prime ideal ${\mathfrak {p}}$ of R.
(iii) P holds for $M_{\mathfrak {m}}$ for all maximal ideal ${\mathfrak {m}}$ of R.

Then the following are local properties in the second sense.

M is zero.
M is torsion-free (when R is a domain)
M is flat.
M is invertible module (when R is a domain and M is a submodule of the field of fractions of R)
$f:M\to N$ is injective (resp. surjective) when N is another R-module.

On the other hand, some properties are not local properties. For example, "noetherian" is (in general) not a local property: that is, to say there is a non-noetherian ring whose localization at every maximal ideal is noetherian: this example is due to Nagata.

Support

The support of the module M is the set of prime ideals p such that M_p ≠ 0. Viewing M as a function from the spectrum of R to R-modules, mapping

p\mapsto M_{p}

this corresponds to the support of a function.

(Quasi-)coherent sheaves

In terms of localization of modules, one can define quasi-coherent sheaves and coherent sheaves on locally ringed spaces. In algebraic geometry, the quasi-coherent O_X-modules for schemes X are those that are locally modelled on sheaves on Spec(R) of localizations of any R-module M. A coherent O_X-module is such a sheaf, locally modelled on a finitely-presented module over R.

Non-commutative case

Localizing non-commutative rings is more difficult; the localization does not exist for every set S of prospective units. One condition which ensures that the localization exists 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.

Notes

↑ Borel, AG. 3.1
↑ Matsumura, Theorem 4.7
↑ Borel, AG. 3.3
↑ Matsumura, a remark after Theorem 4.5

References

Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
Serge Lang, "Algebraic Number Theory," Springer, 2000. pages 3–4.

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