In
algebra, given a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
, there are three ways to change the coefficient ring of a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'',
*
, the induced module.
*
, the coinduced module.
*
, the restriction of scalars.
They are related as
adjoint functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
s:
:
and
:
This is related to
Shapiro's lemma.
Operations
Restriction of scalars
Throughout this section, let
and
be two rings (they may or may not be
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, or contain an
identity), and let
be a homomorphism. Restriction of scalars changes ''S''-modules into ''R''-modules. In
algebraic geometry, the term "restriction of scalars" is often used as a synonym for
Weil restriction.
Definition
Suppose that
is a module over
. Then it can be regarded as a module over
where the action of
is given via
:
where
denotes the action defined by the
-module structure on
.
Interpretation as a functor
Restriction of scalars can be viewed as a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
from
-modules to
-modules. An
-homomorphism
automatically becomes an
-homomorphism between the restrictions of
and
. Indeed, if
and
, then
:
.
As a functor, restriction of scalars is the
right adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
of the extension of scalars functor.
If
is the ring of integers, then this is just the forgetful functor from modules to abelian groups.
Extension of scalars
Extension of scalars changes ''R''-modules into ''S''-modules.
Definition
Let
be a homomorphism between two rings, and let
be a module over
. Consider the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
, where
is regarded as a left
-module via
. Since
is also a right module over itself, and the two actions commute, that is
for
,
(in a more formal language,
is a
-
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...
),
inherits a right action of
. It is given by
for
,
. This module is said to be obtained from
through ''extension of scalars''.
Informally, extension of scalars is "the tensor product of a ring and a module"; more formally, it is a special case of a tensor product of a bimodule and a module – the tensor product of an ''R''-module with an
-bimodule is an ''S''-module.
Examples
One of the simplest examples is
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
, which is extension of scalars from the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s to the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. More generally, given any
field extension ''K'' < ''L,'' one can extend scalars from ''K'' to ''L.'' In the language of fields, a module over a field is called a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, and thus extension of scalars converts a vector space over ''K'' to a vector space over ''L.'' This can also be done for
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
s, as is done in
quaternionification (extension from the reals to the
quaternions).
More generally, given a homomorphism from a field or ''commutative'' ring ''R'' to a ring ''S,'' the ring ''S'' can be thought of as an
associative algebra over ''R,'' and thus when one extends scalars on an ''R''-module, the resulting module can be thought of alternatively as an ''S''-module, or as an ''R''-module with an
algebra representation
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint func ...
of ''S'' (as an ''R''-algebra). For example, the result of complexifying a real vector space (''R'' = R, ''S'' = C) can be interpreted either as a complex vector space (''S''-module) or as a real vector space with a
linear complex structure
In mathematics, a complex structure on a real vector space ''V'' is an automorphism of ''V'' that squares to the minus identity, −''I''. Such a structure on ''V'' allows one to define multiplication by complex scalars in a canonical fashion so ...
(algebra representation of ''S'' as an ''R''-module).
= Applications
=
This generalization is useful even for the study of fields – notably, many algebraic objects associated to a field are not themselves fields, but are instead rings, such as algebras over a field, as in
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
. Just as one can extend scalars on vector spaces, one can also extend scalars on
group algebras and also on modules over group algebras, i.e.,
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s. Particularly useful is relating how
irreducible representations change under extension of scalars – for example, the representation of the cyclic group of order 4, given by rotation of the plane by 90°, is an irreducible 2-dimensional ''real'' representation, but on extension of scalars to the complex numbers, it split into 2 complex representations of dimension 1. This corresponds to the fact that the
characteristic polynomial of this operator,
is irreducible of degree 2 over the reals, but factors into 2 factors of degree 1 over the complex numbers – it has no real eigenvalues, but 2 complex eigenvalues.
Interpretation as a functor
Extension of scalars can be interpreted as a functor from
-modules to
-modules. It sends
to
, as above, and an
-homomorphism
to the
-homomorphism
defined by
.
Co-extension of scalars (coinduced module)
Relation between the extension of scalars and the restriction of scalars
Consider an
-module
and an
-module
. Given a homomorphism
, define
to be the
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
:
,
where the last map is
. This
is an
-homomorphism, and hence
is well-defined, and is a homomorphism (of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s).
In case both
and
have an identity, there is an inverse homomorphism
, which is defined as follows. Let
. Then
is the composition
:
,
where the first map is the
canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
.
This construction shows that the groups
and
are isomorphic. Actually, this isomorphism depends only on the homomorphism
, and so is
functorial
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
. In the language of
category theory, the extension of scalars functor is
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to the restriction of scalars functor.
See also
*
Six operations
*
Tensor product of fields
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...
*
Tensor-hom adjunction In mathematics, the tensor-hom adjunction is that the tensor product - \otimes X and hom-functor \operatorname(X,-) form an adjoint pair:
:\operatorname(Y \otimes X, Z) \cong \operatorname(Y,\operatorname(X,Z)).
This is made more precise below. T ...
References
* {{Cite book, title=Abstract algebra, url=https://archive.org/details/abstractalgebra00dumm_304, url-access=limited, last=Dummit, first=David, date=2004, publisher=Wiley, others=Foote, Richard M., isbn=0471452343, edition=3, location=Hoboken, NJ, oclc=248917264, p
359��377
*J.P. May
Notes on Tor and Ext*
NICOLAS BOURBAKI. Algebra I, Chapter II. LINEAR ALGEBRA.§5. Extension of the ring of scalars;§7. Vector spaces. 1974 by Hermann.
Further reading
Induction and Coinduction of Representations
Commutative algebra
Ring theory
Adjoint functors