Restriction of scalars

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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[edit]

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[edit]

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[edit]

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.