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 one so that it consists of fractions ${\displaystyle {\frac {m}{s}},}$ 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 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 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 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 (c.f. the example given at local ring).

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

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, multiplicative set or multiplicative system. 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 S; as we have supposed S multiplicatively closed, R* is a subring of K. 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 ring homomorphism j : R → R* (as defined above) maps every element of S to a unit in R* = S ⁻¹R. The universal property is that if f : R → T is some other ring homomorphism into another ring T 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.

This can also be phrased in the language of 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.

Examples

• Let R be a commutative ring and f a non-nilpotent element of R. We can consider the multiplicative system {fⁿ : n = 0,1,...}. This localization is obtained precisely by adjoining the root of the polynomial ${\displaystyle ft-1=0}$ in ${\displaystyle R[t]}$ and thus ${\displaystyle S^{-1}R=R[t]/(ft-1)}$. It is typically also denoted as ${\displaystyle R[f^{-1}]}$.
• 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 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. It is a local ring with unique maximal ideal pR_p.
• For the commutative ring ${\displaystyle \mathbb {C} [x,y]}$ its localization at the maximal ideal ${\displaystyle (x,y)}$ is

${\displaystyle \mathbb {C} [x,y]_{(x,y)}=\{f/g:f,g\in \mathbb {C} [x,y]{\text{ and }}g(0,0)\neq 0\}.}$

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, i.e. a ring with one maximal ideal, namely the ideal generated by the extension of P.
• Let R be an integral domain with the field of fractions K. Then its localization ${\displaystyle R_{\mathfrak {p}}}$ at a prime ideal ${\displaystyle {\mathfrak {p}}}$ can be viewed as a subring of K. Moreover,

${\displaystyle R=\bigcap _{\mathfrak {p}}R_{\mathfrak {p}}=\bigcap _{\mathfrak {m}}R_{\mathfrak {m}}}$

where the first intersection is over all prime ideals and the second over the maximal ideals.[1]

• Localization commutes with formations of finite sums, products, intersections and radicals;[2] e.g., if ${\displaystyle {\sqrt {I}}}$ denote the radical of an ideal I in R, then

${\displaystyle {\sqrt {I}}\cdot S^{-1}R={\sqrt {I\cdot S^{-1}R}}\,.}$

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

• The localization can be done element-wise:

${\displaystyle \displaystyle S^{-1}R=\varinjlim R[f^{-1}]}$ where the limit runs over all ${\displaystyle f\in S.}$

Intuition and applications

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.

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 a 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).

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".

Localization of a module

Let R be a commutative ring and S be a multiplicatively closed subset of R (as defined above). 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 the equivalence class of (m, s) by ${\displaystyle {\frac {m}{s}}}$.

To make this set an R-module, define

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

and

${\displaystyle a\cdot {\frac {m}{s}}:={\frac {am}{s}},\ \ \ a\in R.}$

It is straightforward to check that these operations are well-defined, i.e. they give 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 sm/st = m/t for all s,t in S and m in M.

One case is particularly important: if S equals the complement of a prime ideal p ⊂ R (which is multiplicatively closed by definition of a prime ideal) then the localization is denoted M_p instead of (R\p)⁻¹M. 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

${\displaystyle p\mapsto M_{p}}$

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 an R-module M is trivial if and only if all its localizations at primes or maximal ideals are trivial.

Remark:

• There is a module homomorphism

φ: M → S⁻¹M

mapping

φ(m) = m / 1.

Here φ need not be injective, in general, because there may be significant torsion. The additional u showing up in the definition of the above equivalence relation cannot be dropped (otherwise the relation would not be transitive), unless the module is torsion-free.

• By the 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. The corresponding S⁻¹R-module structure is given by ${\displaystyle {\frac {a}{s}}\cdot {\frac {m}{t}}:={\frac {a\cdot m}{st}},}$ where on the right hand side we have a scalar multiplication in the numerator and a ring multiplication in the denominator.

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

Properties

From the definition, one can see that localization of modules is an exact functor, or in other words (reading this in the tensor product) that S⁻¹R is a flat module over R. This fact is foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of the open set Spec(S⁻¹R) into 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

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

is an isomorphism and if ${\displaystyle M}$ is finitely presented, the natural map

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

is an isomorphism.

If a module M is a finitely generated over R,

• ${\displaystyle S^{-1}(\operatorname {Ann} _{R}(M))=\operatorname {Ann} _{S^{-1}R}(S^{-1}M)}$, where ${\displaystyle \operatorname {Ann} }$ denotes annihilator.[4]
• ${\displaystyle S^{-1}M=0}$ if and only if ${\displaystyle tM=0}$ for some ${\displaystyle t\in S}$, which is if and only if ${\displaystyle S}$ intersects the annihilator of ${\displaystyle M}$.[5]

Local property

If ${\displaystyle M}$ is an ${\displaystyle R}$-module, the statement that property P holds for ${\displaystyle M}$ "at a prime ideal ${\displaystyle {\mathfrak {p}}}$" has two possible meanings. The first is that P holds for ${\displaystyle M_{\mathfrak {p}}}$, and the second is that P holds for a neighborhood of ${\displaystyle {\mathfrak {p}}}$. The first interpretation is more common,[6] but for many properties the first and second interpretations coincide. Explicitly, the second means the following conditions are equivalent:

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

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 (when R is a domain and M is a submodule of the field of fractions of R).
• ${\displaystyle f:M\to N}$ 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.

(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. 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.

See also

• Completion (algebra)
• Homomorphism
• Overring
• Valuation ring
• Amitsur complex, a generalization of a localization

Localization

Category:Localization (mathematics)

• Local analysis
• Localization of a category
• Localization of a topological space

References

1. Matsumura, Theorem 4.7
2. Atiyah & MacDonald 1969, Proposition 3.11. (v). harvnb error: no target: CITEREFAtiyahMacDonald1969 (help)
3. Borel, AG. 3.3
4. Atiyah & MacDonald, Proposition 3.14. harvnb error: no target: CITEREFAtiyahMacDonald (help)
5. Borel, AG. 3.1
6. Matsumura, a remark after Theorem 4.5

• Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
• Cohn, P. M. (1989). "§ 9.3". Algebra. Vol. 2 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xvi+428. ISBN 0-471-92234-X. MR 1006872.
• Cohn, P. M. (1991). "§ 9.1". Algebra. Vol. 3 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xii+474. ISBN 0-471-92840-2. MR 1098018.
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960
• 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.