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 $f:S\to R$ is the operation on $R$-modules given by forming the tensor product of modules with $R$ regarded as an $S$-module via $f$.

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

Definition

Let $R$ and $S$ be commutative rings and let $f:R\to S$ 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

$$f^{*}:S\mathrm{Mod}\to R\mathrm{Mod}$$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

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

Proposition

The restriction of scalars functor, def. 1, is the right adjoint in a pair of adjoint functors

$$(f_{!}⊣f^{*}):S\mathrm{Mod}\stackrel{\stackrel{f_{!}}{\leftarrow }}{\underset{f^{*}}{\to }}R\mathrm{Mod}.$$( f_! \dashv f^* ) : S Mod \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} R Mod \,.

Definition

The left adjoint $f_{!}:R\mathrm{Mod}\to S\mathrm{Mod}$ in prop. 1 is called extension of scalars along $f$.

Remark

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

In components

Proposition

Given a ring homomorphism $f:R\to S$, the extension of scalars functor $f_{!}$ of def. 2 is the functor

$$f_{!}≔S{\otimes }_R(-):R\mathrm{Mod}\to S\mathrm{Mod}$$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 $N\in S\mathrm{Mod}$

• 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 $s\prime \cdot (s,n)=(s\prime \cdot s,n)$.

Properties

Geometric interpretation

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

By definition the category

$${\mathrm{Ring}}^{\mathrm{op}}\underset{\mathrm{Spec}}{\overset{:\simeq }{\to }}\mathrm{Aff}$$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

$$\mathrm{Spec}(f):\mathrm{Spec}(S)\to \mathrm{Spec}(R)$$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 $\mathrm{Spec}(R)$: something that has a quasicoherent sheaf of sections.

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

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

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

Examples

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

• localization of a module

Related concepts

• coextension of scalars