Localization (commutative algebra)

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, 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.

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.

• 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

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

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 ${\displaystyle f\in S.}$

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]

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: Consider the boolean ring ${\displaystyle R=\prod _{i=1}^{\infty }\mathbb {Z} /\mathbb {2Z} }$. Then ${\displaystyle R}$ is not noetherian because a boolean noetherian ring must be finite. However, a local boolean ring is a field isomorphic to ${\displaystyle \mathbb {Z} /\mathbb {2Z} }$, hence noetherian.

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.

Category:Localization (mathematics)

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