symmetric monoidal (∞,1)-category of spectra
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.
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.
Write $R$Mod and $S$Mod for the categories of modules over $R$ and $S$, respectively.
Given a ring homomorphism $f : R \to S$ the restriction of scalars functor
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
The restriction of scalars functor, def. 1, is the right adjoint in a pair of adjoint functors
The left adjoint $f_! \colon R Mod \to S Mod$ in prop. 1 is called extension of scalars along $f$.
A further right adjoint $f_*$ would be called coextension of scalars along $f$.
Given a ring homomorphism $f : R \to S$, the extension of scalars functor $f_!$ of def. 2 is the functor
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 $N \in S Mod$
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 $s' \cdot(s,n) = (s' \cdot s,n)$.
Under Isbell duality extension of scalars turns into a statement about geometry.
By definition the category
of (absolute) affine schemes is the opposite category of Ring.
Hence for $f : R \to S$ a ring homomorphism, we have equivalently a morphism
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.
complexification is extension of scalars along the inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$ of the real numbers into the complex numbers.