In algebra, given a ring homomorphism
Contents
- Restriction of scalars
- Definition
- Interpretation as a functor
- The case of fields
- Extension of scalars
- Examples
- Relation between the extension of scalars and the restriction of scalars
- References
They are related as adjoint functors:
and
This is related to Shapiro's lemma.
Restriction of scalars
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
Let
Interpretation as a functor
Restriction of scalars can be viewed as a functor from
As a functor, restriction of scalars is the right adjoint of the extension of scalars functor.
If
The case of fields
When both
Extension of scalars
Extension of scalars changes R-modules into S-modules.
Definition
In this definition the rings are assumed to be associative, but not necessarily commutative, or to have an identity. Also, modules are assumed to be left modules. The modifications needed in the case of right modules are straightforward.
Let
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
Examples
One of the simplest examples is complexification, which is extension of scalars from the real numbers to the complex numbers. 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, and thus extension of scalars converts a vector space over K to a vector space over L. This can also be done for division algebras, 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 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 (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. 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 representations. 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,
Interpretation as a functor
Extension of scalars can be interpreted as a functor from
Relation between the extension of scalars and the restriction of scalars
Consider an
where the last map is
In case both
where the first map is the canonical isomorphism
This construction shows that the groups