Localization (algebra)

In algebra , localization is a method of systematically adding new multiplicative inverse elements to a ring . If you want the elements of a subset of to be invertible, you construct a new ring , the “localization from to ”, and a ring homomorphism from to , which maps to units of . and this ring homomorphism satisfies the universal property of "best choice". ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle S^{-1}R}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle S^{-1}R}$ ${\displaystyle S}$ ${\displaystyle S^{-1}R}$ ${\displaystyle S^{-1}R}$

In this article we limit ourselves to commutative rings with one element 1. In the case of a ring without one element, questions of invertibility do not arise or only after the adjunct of a single element . For a generalization to the case of non-commutative rings see the Ore condition .

Word origin

The use of the term “localization” comes from algebraic geometry : If a ring of real or complex-valued functions on a geometric object (e.g. an algebraic variety ) and if one wants to investigate the behavior of the functions in the vicinity of a point , then choose one for the set of functions that are not equal to 0 and localizes to . The localization then only contains information about the behavior of the functions close by . ${\displaystyle R}$ ${\displaystyle p}$ ${\displaystyle S}$ ${\displaystyle p}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle p}$

definition

Localization of a general commutative ring

Let be a commutative ring with 1 and a subset of . Since the product of units is again a unit, 1 is a unit and we want to make the elements of into units, we can enlarge and add the 1 and all products of elements of to ; so we immediately assume that multiplicative is closed and contains the unity. We then introduce an equivalence relation on the Cartesian product : ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle R\times S}$

${\displaystyle (r_{1},s_{1})\sim (r_{2},s_{2}):\Leftrightarrow \exists t\in S:t(r_{1}s_{2}-r_{2}s_{1})=0}$.

The factor appearing in the definition of the equivalence relation is necessary for the transitivity of the equivalence relation if the present ring is not zero divisor free . The equivalence class of a couple we write as a fraction ${\displaystyle t}$ ${\displaystyle (r_{1},s_{1})}$

${\displaystyle {\frac {r_{1}}{s_{1}}}:=[(r_{1},s_{1})]:=\{(r_{2},s_{2})\in R\times S:(r_{1},s_{1})\sim (r_{2},s_{2})\}}$.

Addition and multiplication of the equivalence classes are defined analogously to the usual fractions calculation rules (the well-definedness, i.e. the independence from the choice of the special representative , must be shown):

${\displaystyle {\frac {r_{1}}{s_{1}}}+{\frac {r_{2}}{s_{2}}}:={\frac {r_{1}s_{2}+r_{2}s_{1}}{s_{1}s_{2}}}}$

${\displaystyle {\frac {r_{1}}{s_{1}}}\cdot {\frac {r_{2}}{s_{2}}}:={\frac {r_{1}r_{2}}{s_{1}s_{2}}}}$

With the links defined in this way, we get a ring . The image ${\displaystyle S^{-1}R}$

${\displaystyle j\colon R\to S^{-1}R,\ j(r)={\frac {rs}{s}}}$

with is a (not necessarily injective) ring homomorphism and is independent of the choice of . ${\displaystyle s\in S}$ ${\displaystyle s}$

Localization of an integrity ring

In the simplest case it is an integrity ring . Here we differentiate whether the contains 0 or not. ${\displaystyle R}$ ${\displaystyle S}$

If , then only the zero ring comes into question for the localization , because it is the only ring in which the 0 unit is located. So we define if 0 is in . ${\displaystyle 0\in S}$ ${\displaystyle \{0\}}$ ${\displaystyle S^{-1}R=\{0\}}$ ${\displaystyle S}$

If 0 is not an element of , then the above equivalence relation is simplified, since because of the law of abbreviation valid in integrity rings it follows: if and only if . Special is also closed multiplicatively, and the above construction coincides with the known construction of the quotient field . ${\displaystyle S}$ ${\displaystyle (r_{1},s_{1})\,\sim \,(r_{2},s_{2})}$ ${\displaystyle r_{1}s_{2}=r_{2}s_{1}}$ ${\displaystyle R\setminus \{0\}}$

Localizations after a multiplicatively closed subset can then be found in the quotient field of as follows . The partial ring of , which consists of all fractions whose numerator is in and whose denominator is in , has the desired properties: The canonical embedding of in is a ring homomorphism that is even injective , and the elements of are invertible. This ring is the smallest partial ring of that contains and in which the elements of are invertible. ${\displaystyle S\subset R\setminus \{0\}}$ ${\displaystyle K}$ ${\displaystyle R}$ ${\displaystyle S^{-1}R}$ ${\displaystyle K}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle K}$ ${\displaystyle S}$ ${\displaystyle S^{-1}R}$ ${\displaystyle K}$ ${\displaystyle R}$ ${\displaystyle S}$

Here are some examples of localizations for different subsets : ${\displaystyle \mathbb {Z} }$ ${\displaystyle S}$

• If you localize with respect to the set of odd whole numbers, you get the ring of all rational numbers with an odd denominator. The use of “(2)” is explained below. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} _{(2)}}$
• If you localize with respect to the set of even numbers without the 0, you get whole , because every rational number can be represented as a fraction with an even denominator through possible expansion with 2. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} }$
• If you localize with respect to the set of powers of two, you get the ring of dual fractions. These are exactly the rational numbers whose dual representation only has a finite number of decimal places . ${\displaystyle \mathbb {Z} }$

Category theoretical definition

The localization of a ring according to a subset can be defined in terms of category theory as follows : ${\displaystyle R}$ ${\displaystyle S}$

If R is a ring and a subset, then the set of all -algebras that are such that under the canonical injection each element of is mapped to a unit , forms a category with -algebra homomorphisms as morphisms. The localization from to is then the initial object of this category. ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R\to A}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle S}$

This corresponds to the definition given above, which is more easily accessible algebraically, as is usually found in textbooks on commutative algebra.

Universal property

The "best choice" of the ring and homomorphism is defined by the fulfillment of a universal property : ${\displaystyle S^{-1}R}$ ${\displaystyle j\colon R\to S^{-1}R}$

If a commutative ring with 1, a multiplicatively closed subset of , a ring with 1, a ring homomorphism, which maps each element of to a unit, then there is exactly one ring homomorphism with . ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle T}$ ${\displaystyle t\colon R\to T}$ ${\displaystyle S}$ ${\displaystyle g\colon S^{-1}R\to T}$ ${\displaystyle t=g\circ j}$

This corresponds to the category-theoretical definition as an initial object. The algebraic construction given above is a ring for which this universal property can be demonstrated.

Common types of localization

Localization on an element

By betting, one allows all powers of an element as denominators. Common spellings for this are , or . The localization obtained is canonically isomorphic to , the isomorphism fix point by point and to reflect (or vice versa). For example, the ring of Laurent polynomials is created in this way. ${\displaystyle S:=\{r^{n}\mid n\in \mathbb {N} _{0}\}}$ ${\displaystyle r\in R}$ ${\displaystyle R_{r}}$ ${\displaystyle R\left[{\tfrac {1}{r}}\right]}$ ${\displaystyle R[r^{-1}]}$ ${\displaystyle R[X]/(rX-1)}$ ${\displaystyle R}$ ${\displaystyle {\tfrac {1}{r}}}$ ${\displaystyle X}$

Localization according to a prime ideal

When a prime ideal denotes, one speaks for of the "localization in " or "after " . The resulting ring is local with the maximum ideal . Is precisely the above mentioned ring homomorphism, then , an inclusion-preserving bijection. The above ring for a prime number is an example of this construction. ${\displaystyle {\mathfrak {p}}\in \operatorname {Spec} (R)}$ ${\displaystyle S:=R\setminus {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle R_{\mathfrak {p}}:=S^{-1}R}$ ${\displaystyle R_{\mathfrak {p}}{\mathfrak {p}}:=S^{-1}{\mathfrak {p}}}$ ${\displaystyle j\colon R\to R_{\mathfrak {p}}}$ ${\displaystyle \operatorname {Spec} (R_{\mathfrak {p}})\to \{{\mathfrak {a}}\in \operatorname {Spec} (R)|{\mathfrak {a}}\subseteq {\mathfrak {p}}\}}$ ${\displaystyle {\mathfrak {b}}\mapsto j^{-1}({\mathfrak {b}})}$ ${\displaystyle \mathbb {Z} _{(p)}}$ ${\displaystyle p}$

Since there is no zero divisor, the quotient field can be formed. It then applies . ${\displaystyle R/{\mathfrak {p}}}$ ${\displaystyle \operatorname {Quot} (R/{\mathfrak {p}})\cong R_{\mathfrak {p}}/R_{\mathfrak {p}}{\mathfrak {p}}}$

The localization according to a prime ideal can also be interpreted as follows: If one understands elements of as functions on the spectrum of , whose value at one point is the respective image in the remainder class field , then the local ring "consists" of fractions in their denominators There are functions that do not disappear with, “which can be shared locally with ”. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle P}$ ${\displaystyle \kappa (P):=R_{\mathfrak {p}}/R_{\mathfrak {p}}{\mathfrak {p}}}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P}$

" Completely closed " is a local property; H. for a zero-divisor-free ring are equivalent: ${\displaystyle R}$

• ${\displaystyle R}$ is completely closed
• ${\displaystyle R_{\mathfrak {p}}}$ is completely closed for all prime ideals ${\displaystyle {\mathfrak {p}}\triangleleft R}$
• ${\displaystyle R_{\mathfrak {m}}}$ is completely closed for all maximum ideals ${\displaystyle {\mathfrak {m}}\triangleleft R}$

Total quotient ring

The total quotient ring of a ring is the localization of at the set of non-zero divisors of . It is the "strongest" isolation for which the isolation mapping ${\displaystyle Q}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle R\to Q,\quad r\mapsto r/1}$

is injective . If there is an integrity ring , the total quotient ring is the quotient field of . ${\displaystyle R}$ ${\displaystyle R}$

Ideal theory of localization

Let it be a commutative ring and be closed multiplicatively. It denotes the canonical ring homomorphism . ${\displaystyle R}$ ${\displaystyle S\subseteq R}$ ${\displaystyle f\colon R\longrightarrow S^{-1}R}$

Then holds for any ideal ${\displaystyle I\subseteq S^{-1}R}$

${\displaystyle I=f_{*}(f^{*}(I))}$

In particular, then, every ideal of is the image of an ideal of . ${\displaystyle S^{-1}R}$ ${\displaystyle R}$

Prime ideals

The image

${\displaystyle f^{*}\colon \mathrm {Spec} (S^{-1}R){\overset {\sim }{\longrightarrow }}{\Big \{}{\mathfrak {p}}\in \mathrm {Spec} (R)\,\,{\Big |}\,\,{\mathfrak {p}}\cap S=\emptyset {\Big \}}}$

is bijective. The prime ideals of localization are therefore precisely the images (below ) of the prime ideals of , which have no element in common with the set . ${\displaystyle f}$ ${\displaystyle R}$ ${\displaystyle S}$

The localization according to a prime ideal provides a ring that only has one maximum ideal (the image of ). This makes the ring a local ring with a maximum ideal , which justifies the name localization . On the other hand, there can be several prime ideals in localization, for example in the localization of an integrity area, which is itself also an integrity area, the zero ideal. Further prime ideals can be excluded if it is at most one-dimensional or especially a Dedekind area. ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle f_{*}({\mathfrak {p}})}$ ${\displaystyle R}$

Localization of modules

Is a commutative ring having 1, a multiplicative subset of , and a module, so that is localization of relative defined as the amount of the equivalence classes of pairs , also written , said two pairs , should be equivalent if there is an element of are so that ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle S}$ ${\displaystyle S^{-1}M}$ ${\displaystyle (m,s)}$ ${\displaystyle m/s}$ ${\displaystyle \left(m_{1},s_{1}\right)}$ ${\displaystyle \left(m_{2},s_{2}\right)}$ ${\displaystyle s}$ ${\displaystyle S}$

${\displaystyle s\left(s_{2}\cdot m_{1}-s_{1}\cdot m_{2}\right)=0}$

applies. is a module. ${\displaystyle S^{-1}M}$ ${\displaystyle S^{-1}R}$

Corresponding to the case of rings, one also writes or for elements or maximum ideals of . ${\displaystyle M_{r}}$ ${\displaystyle M_{P}}$ ${\displaystyle r}$ ${\displaystyle P}$ ${\displaystyle R}$

The localization of a module also has a universal property: Any homomorphism of into a module in which all elements are "divisible" by the elements of , i. H. the left multiplication with an element of a module isomorphism can be unequivocally continued to a homomorphism . This means that the localization of a module can also be described as a tensor product : ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle S^{-1}M\to N}$

${\displaystyle S^{-1}M=M\otimes _{R}S^{-1}R}$.

properties

Let be a commutative ring and , two -modules, as well as multiplicatively closed. Then applies ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle S\subseteq R}$

• ${\displaystyle S^{-1}M{\Big /}S^{-1}N=S^{-1}{\Big (}M/N{\Big )}}$
• ${\displaystyle S^{-1}M\bigotimes \nolimits _{R}S^{-1}N=S^{-1}{\Big (}M\bigotimes \nolimits _{R}N{\Big )}}$
• For -Untermoduln applies: . The statement is generally wrong for infinite cuts. ${\displaystyle R}$ ${\displaystyle M_{1},\dotsc ,M_{n}}$ ${\displaystyle S^{-1}\bigcap \limits _{i=1}^{n}M_{i}=\bigcap \limits _{i=1}^{n}S^{-1}M_{i}}$
  
• There is also a criterion for the localization of a finitely generated module, when the localization delivers the null module :

${\displaystyle S^{-1}M=0\quad \Longleftrightarrow \quad \exists s\in S:\quad sM=0}$

So the localization is zero if and only if an element that cancels the whole module is contained in the set . ${\displaystyle S}$

In the case of an infinitely generated module, this criterion no longer applies.

See also

• A very similar construction is the so-called Grothendieck group , which constructs a commutative group from a commutative semigroup by adding suitable inverses.
• The localization of a non-commutative ring is much more difficult and not always possible. One possible generalization to non-commutative rings is provided by the Ore conditions .

swell

• Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Atiyah, Macdonald: Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9