nLab extension of scalars

Idea

The extension of scalars of a module along a homomorphism of rings is the algebraic dual of what geometrically is the pullback of bundles along a map of their base spaces (with respect to the discussion at modules - as generalized vector bundles).

Explicitly, extension of scalars along a ring homomorphism $f : R \to S$ is the operation on $R$-modules given by forming the tensor product of modules with $S$ regarded as an $R$-module via $f$.

There are similar functors for bimodules and in some other categories.

Definition

Let $R$ and $S$ be commutative rings and let $f \colon R\to S$ be a homomorphism of rings.

We discuss extension of scalars along $f$ first general abstractly and then explicitly in components.

General abstract

Write $R$Mod and $S$Mod for the categories of modules over $R$ and $S$, respectively.

Definition

Given a ring homomorphism $f : R \to S$ the restriction of scalars functor

$$f^* : S Mod \to R Mod$$

is the functor that takes an $S$-module $N$ to the $R$-module $f^*N$ whose underlying abelian group is that of $N$ and whose $R$-action is given by

$$r \cdot n \coloneqq f(r)\cdot n \;\;\;\; for \; r \in R, n \in N \,.$$

Proposition

The restriction of scalars functor, def. , is the right adjoint in a pair of adjoint functors

$$( f_! \dashv f^* ) \;\colon\; S Mod \stackrel{\overset{f_!}{\longleftarrow}} {\underset{f^*}{\longrightarrow}} R Mod \,.$$

Definition

The left adjoint $f_! \colon R Mod \to S Mod$ in prop. is called extension of scalars along $f$.

Remark

A further right adjoint $f_*$ would be called coextension of scalars along $f$.

See also induced representation $\dashv$ restricted representation.

In components

Proposition

Given a ring homomorphism $f : R \to S$, the extension of scalars functor $f_!$ of def. is the functor

$$f_! \coloneqq S \otimes_R (-) \,:\, R Mod \to S Mod$$

given by tensor product of modules with $S$ regarded as an $S$-$R$-bimodule: the left action being the canonical action of $S$ on itself, the right being the restriction of scalars-action along $f$.

Explicitly, for $N \in R Mod$

• the elements of $f_! N$ are equivalence classes of pairs $(s,n) \in S \times N$ under the equivalence relation $(s \cdot f(r), n) = (s, r\cdot n)$ for all $s \in S$;

• the left $S$-action is given by $s' \cdot(s,n) = (s' \cdot s,n)$.

Properties

Geometric interpretation

Under Isbell duality extension of scalars turns into a statement about geometry.

By definition the category

$$Ring^{op} \underoverset{Spec}{\colon \simeq}{\to} Aff$$

of (absolute) affine schemes is the opposite category of Ring.

Hence for $f : R \to S$ a ring homomorphism, we have equivalently a morphism

$$Spec(f) : Spec(S) \to Spec(R)$$

of affine schemes.

An $R$-module $N$ corresponds to the collection of sections of a “generalized vector bundle” over $Spec(R)$: something that has a quasicoherent sheaf of sections.

The pullback of this “bundle” along $Spec(f)$ has sections forming the module $f_! N$.

Generally, for any fibered category like Mod$\to Aff$ we may regard the inverse image functor as the extension of scalars.

For that reason if there is some other fibered category $\mathcal{F}$ over the opposite of some algebraic category $\mathcal{A}$ whose objects are considered “objects of scalars” one is inclined to call the inverse image functor, the extension of scalars.

Frobenius extensions

Extension of scalars coincides with coextension of scalars (to make an ambidextrous adjunction with restriction of scalars) in the case of Frobenius extensions, see there for more.

Examples

• complexification is extension of scalars along the inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$ of the real numbers into the complex numbers.

• localization of a module

Related concepts

• extension of scalars $\dashv$ restriction of scalars $\dashv$ coextension of scalars

• extension (double category theory)