Restriction of scalars

In abstract algebra, restriction of scalars is a procedure of creating a module over a ring $R$ from a module over another ring $S$ , given a homomorphism $f : R \to S$ between them. Intuitively speaking, the resulting module "remembers" less information than the initial one, hence the name.

In algebraic geometry, the term "restriction of scalars" is often used as a synonym for Weil restriction.

Definition

Let $R$ and $S$ be two rings (they may or may not be commutative, or contain an identity), and let $f:R \to S$ be a homomorphism. Suppose that $M$ is a module over $S$ . Then it can be regarded as a module over $R$ , if the action of $R$ is given via $r \cdot m = f(r) \cdot m$ for $r \in R$ and $m \in M$ .

Interpretation as a functor

Restriction of scalars can be viewed as a functor from $S$ -modules to $R$ -modules. An $S$ -homomorphism $u : M \to N$ automatically becomes an $R$ -homomorphism between the restrictions of $M$ and $N$ . Indeed, if $m \in M$ and $r \in R$ , then

u(r \cdot m) = u(f(r) \cdot m) = f(r) \cdot u(m) = r\cdot u(m)\,

As a functor, restriction of scalars is the right adjoint of the extension of scalars functor.

If $R$ is the ring of integers, then this is just the forgetful functor from modules to abelian groups.

The case of fields

When both $R$ and $S$ are fields, $f\$ is necessarily a monomorphism, and so identifies $R$ with a subfield of $S$ . In such a case an $S$ -module is simply a vector space over $S$ , and naturally over any subfield thereof. The module obtained by restriction is then simply a vector space over the subfield $R \subset S$ .