Algebras and modules
Model category presentations
Geometry on formal duals of algebras
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 is the operation on -modules given by forming the tensor product of modules with regarded as an -module via .
There are similar functors for bimodules and in some other categories.
Let and be commutative rings and let be a homomorphism of rings.
We discuss extension of scalars along first general abstractly and then explicitly in components.
Write Mod and Mod for the categories of modules over and , respectively.
Given a ring homomorphism the restriction of scalars functor
is the functor that takes an -module to the -module whose underlying abelian group is that of and whose -action is given by
The left adjoint in prop. 1 is called extension of scalars along .
Given a ring homomorphism , the extension of scalars functor of def. 2 is the functor
given by tensor product of modules with regarded as an --bimodule: the left action being the canonical action of on itself, the right being the restriction of scalars-action along .
the elements of are equivalence classes of pairs under the equivalence relation for all ;
the left -action is given by .
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 a ring homomorphism, we have equivalently a morphism
of affine schemes.
An -module corresponds to the collection of sections of a “generalized vector bundle” over : something that has a quasicoherent sheaf of sections.
The pullback of this “bundle” along has sections forming the module .
Generally, for any fibered category like Mod we may regard the inverse image functor as the extension of scalars.
For that reason if there is some other fibered category over the opposite of some algebraic category whose objects are considered “objects of scalars” one is inclined to call the inverse image functor, the extension of scalars.