Localization of a module
Encyclopedia
In algebraic geometry
, the localization of a module is a construction to introduce denominators in a module
for a ring
. More precisely, it is a systematic way to construct a new module S−1M out of a given module M containing algebraic fractions
.
where the denominators s range in a given subset S of R.
The technique has become fundamental, particularly in algebraic geometry
, as the link between modules and sheaf
theory. Localization of a module generalizes localization of a ring
.
and M an R-module
.
Let S a multiplicatively closed subset of R, i.e. for any s and t ∈ S, the product st is also in S. 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
It is common to denote these equivalence classes
.
To make this set a R-module, define
and
(a ∈ R). It is straightforward to check that the definition is well-defined, i.e. yields 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 rs/us = r/u for all s in S.
One case is particularly important: 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 Mp instead of (R\p)−1M. The support of the module M is the set of prime ideals p such that Mp ≠ 0. Viewing M as a function from the spectrum
of R to R-modules, mapping
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 a R-module M is trivial if and only if all its localizations at primes or maximal ideals are trivial.
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
.
, 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
.
on locally ringed spaces. In algebraic geometry, the quasi-coherent OX-modules for scheme
s X are those that are locally modelled on sheaves on Spec(R) of localizations of any R-module M. A coherent OX-module is such a sheaf, locally modelled on a finitely-presented module over R.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the localization of a module is a construction to introduce denominators in a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
for a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. More precisely, it is a systematic way to construct a new module S−1M out of a given module M containing algebraic fractions
.where the denominators s range in a given subset S of R.
The technique has become fundamental, particularly in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, as the link between modules and sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
theory. Localization of a module generalizes localization of a ring
Localization of a ring
In abstract 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 in R*...
.
Definition
In this article, let R be a commutative ringCommutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
and M an R-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
.
Let S a multiplicatively closed subset of R, i.e. for any s and t ∈ S, the product st is also in S. 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
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
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 these equivalence classes
.To make this set a R-module, define
and
(a ∈ R). It is straightforward to check that the definition is well-defined, i.e. yields 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 rs/us = r/u for all s in S.
One case is particularly important: if S equals the complement of a prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
p ⊂ R (which is multiplicatively closed by definition of prime ideals) then the localization is denoted Mp instead of (R\p)−1M. The support of the module M is the set of prime ideals p such that Mp ≠ 0. Viewing M as a function from the spectrum
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
of R to R-modules, mapping
this corresponds to the support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
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 a R-module M is trivial if and only if all its localizations at primes or maximal ideals are trivial.
Remarks
- The definition applies in particular to M=R, and we get back the localized ringLocalization of a ringIn abstract 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 in R*...
S−1R.
- 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 torsionTorsionThe word torsion may refer to the following:*In geometry:** Torsion of a curve** Torsion tensor in differential geometry** The closely related concepts of Reidemeister torsion and analytic torsion ** Whitehead torsion*In algebra:** Torsion ** Tor functor* In medicine:** Ovarian...
. The additional u showing up in the definition of the above equivalence relation can not be dropped (otherwise the relation would not be transitive), unless the module is torsion-free.
- Some authors allow not necessarily multiplicatively closed sets S and define localizations in this situation, too. However, saturating such a set, i.e. adding 1 and finite products of all elements, this comes down to the above definition.
Tensor product interpretation
By the very definitions, the localization of the module is tightly linked to the one of the ring via the tensor productTensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
- S−1M = M ⊗RS−1R,
This way of thinking about localising is often referred to as extension of scalars
Extension of scalars
In abstract algebra, extension of scalars is a means of producing a module over a ring S from a module over another ring R, given a homomorphism f : R \to S between them...
.
As a tensor product, the localization satisfies the usual universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
.
Flatness
From the definition, one can see that localization of modules is an exact functorExact functor
In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
, or in other words (reading this in the tensor product) that S−1R is a flat module
Flat module
In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...
over R. This is actually foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
in Spec(R) (see spectrum of a ring
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
) is a flat morphism
Flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,is a flat map for all P in X...
.
(Quasi-)coherent sheaves
In terms of localization of modules, one can define quasi-coherent sheaves and coherent sheavesCoherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
on locally ringed spaces. In algebraic geometry, the quasi-coherent OX-modules for scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s X are those that are locally modelled on sheaves on Spec(R) of localizations of any R-module M. A coherent OX-module is such a sheaf, locally modelled on a finitely-presented module over R.