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extension of scalars

Contents

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

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

Definition

Let RR and SS be commutative rings and let f:RSf \colon R\to S be a homomorphism of rings.

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

General abstract

Write RRMod and SSMod for the categories of modules over RR and SS, respectively.

Definition

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

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

is the functor that takes an SS-module NN to the RR-module f *Nf^*N whose underlying abelian group is that of NN and whose RR-action is given by

rnf(r)nforrR,nN. 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 *):SModf *f !RMod. ( f_! \dashv f^* ) : S Mod \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} R Mod \,.
Definition

The left adjoint f !:RModSModf_! \colon R Mod \to S Mod in prop. 1 is called extension of scalars along ff.

Remark

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

In components

Proposition

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

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

given by tensor product of modules with SS regarded as an SS-RR-bimodule: the left action being the canonical action of SS on itself, the right being the restriction of scalars-action along ff.

Explicitly, for NSModN \in S Mod

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

  • the left SS-action is given by s(s,n)=(ss,n)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 opSpec:Aff Ring^{op} \underoverset{Spec}{\colon \simeq}{\to} Aff

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

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

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

of affine schemes.

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

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

Generally, for any fibered category like ModAff\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 𝒜\mathcal{A} whose objects are considered “objects of scalars” one is inclined to call the inverse image functor, the extension of scalars.

Examples

Revised on September 11, 2013 14:07:50 by Igor Khavkine (82.157.45.36)