Given a ring map R→S, the S-module S⊗_R M obtained from an R-module M by base change.
Let f:R→S be a homomorphism of commutative rings, and let M be an R-module. The extension of scalars (or base change) of M along f is the S-module
S⊗RM,
where S acts on the left tensor factor: s⋅(s′⊗m)=(ss′)⊗m.
There is a canonical R-linear map
ηM:M⟶S⊗RM,m⟼1⊗m,
where S⊗RM is viewed as an R-module via f.
Universal property and adjunction
For every S-module N, restriction of scalars along f produces an R-module; this is restriction of scalars. Extension of scalars is left adjoint to restriction of scalars, meaning there is a natural bijection
HomS(S⊗RM,N)≅HomR(M,ResfN),
where ResfN denotes N viewed as an R-module via f.
Quotient base change. Let S=R/I and f:R→R/I be the quotient map. Then for any R-module M,
(R/I)⊗RM≅M/IM.
For example, with R=Z, S=Z/nZ, one gets (Z/n)⊗ZM≅M/nM.
Field extension. If k⊆K is a field extension and V is a k-vector space, then K⊗kV is the K-vector space obtained by extending scalars. If V≅kd is finite-dimensional, then K⊗kV≅Kd.
Localization as extension of scalars. Let R=k[x], let S={1,x,x2,…}, and set R′=S−1R≅k[x,x−1]. For M=R/(x), extension of scalars gives
R′⊗RM≅S−1M=0,
since x becomes invertible after localization but kills M.
A commutative ring is a ringR such that ab=ba for all a,b∈R.
Let R and S be commutative rings, and let φ:R→S be a ring homomorphism. If M is an S-module, the restriction of scalars of M along φ is the R-module
Resφ(M),
defined as follows:
As an abelian group, Resφ(M) is the same underlying abelian group as M.
The R-action is given by
r⋅m:=φ(r)mfor r∈R,m∈M,
where the multiplication on the right is the original S-module structure on M.
This construction is functorial in M and defines a forgetful functor
Resφ:ModS⟶ModR.
It is faithful and exact (it does not change the underlying abelian groups or group homomorphisms).
Restriction of scalars is the natural companion to extension of scalars, and in many settings these form an adjoint pair (extension is left adjoint to restriction).
Examples
From a polynomial algebra to the base field. Let k be a field and φ:k↪k[x] the usual inclusion. If M=k[x] is viewed as a k[x]-module over itself, then Resφ(M) is just the underlying k-vector space of polynomials, which is infinite-dimensional over k.
Forgetting an S-module to a Z-module. For the canonical surjection φ:Z→Z/nZ, any Z/nZ-module M becomes a Z-module by restriction. Concretely, Resφ(M) is the underlying abelian group of M, and it satisfies nM=0, i.e. n lies in the annihilator of Resφ(M).
Restriction along localization. If S=Rf is the localization of R at a single element f∈R, then every Rf-module restricts to an R-module along R→Rf. For instance, Rf itself (as an Rf-module) becomes an R-module in which multiplication by f is invertible; compare this with localization of modules.
As a set, S−1R can be constructed from pairs (r,s)∈R×S modulo the equivalence relation
(r,s)∼(r′,s′)⟺∃t∈S such that t(rs′−r′s)=0 in R.
Write the class of (r,s) as sr. Addition and multiplication are defined by
sr+s′r′=ss′rs′+r′s,sr⋅s′r′=ss′rr′.
The canonical map is ι(r)=1r.
If 0∈S, then ι(0) is invertible, hence 1=0 in S−1R; in this case S−1R is the zero ring.
Universal property
The localization is characterized by the following universal mapping property:
If A is any commutative ring and φ:R→A is a ring homomorphism such that φ(s) is a unit of A for every s∈S, then there exists a unique ring homomorphism φ:S−1R→A with φ∘ι=φ. Explicitly,
Let R be a commutative ring. A subset S⊆R is a multiplicative set if
1∈S, and
whenever s,t∈S, then st∈S.
Often one also assumes 0∈/S when the goal is to form the localization of a ringS−1R; if 0∈S, then 0 becomes invertible in S−1R, forcing 1=0 and hence S−1R is the zero ring.
A key source of multiplicative sets is complements of primes: if p⊂R is prime, then R∖p is multiplicative, and this choice produces the localization at a prime.
Examples
Powers of an element. For f∈R, the set
S={1,f,f2,f3,…}
is multiplicative. (If f is nilpotent, then 0∈S and the corresponding localization collapses to the zero ring.)
Complement of a prime ideal. If p is a prime ideal of R, then
S=R∖p
is multiplicative (primality ensures st∈/p whenever s,t∈/p). Localizing at this S gives Rp.
Inverting a prime number in Z. In R=Z, the subset S={1,p,p2,…} (for a prime p) is multiplicative. The localization S−1Z is the subring of Q consisting of fractions whose denominator is a power of p.
Let R be a commutative ring, let S⊆R be a multiplicative set, and let M be an R-module. The localization of M at S is an S−1R-module, denoted S−1M, constructed so that every s∈S acts invertibly on S−1M.
Define S−1M as equivalence classes of pairs (m,s)∈M×S under
(m,s)∼(m′,s′)⟺∃t∈S such that t(s′m−sm′)=0 in M.
Write the class of (m,s) as sm. Addition is
sm+s′m′=ss′s′m+sm′,
and the scalar action of S−1R is given by
(sr)(tm)=strm.
The map ιM:M→S−1M given by ιM(m)=1m is R-linear.
Universal property
Let N be an S−1R-module. Viewing N as an R-module via the canonical map R→S−1R, every s∈S acts by an automorphism on N. The localization S−1M is characterized by:
For every R-linear map f:M→N, there exists a unique S−1R-linear map f:S−1M→N with f∘ιM=f.
In particular, localizing at a prime p means taking S=R∖p and writing
Localization interacts well with exact sequences: it is an exact functor on modules (see exactness of localization and compare with the general notion of an exact sequence).
Finally, localization can be expressed as a base change: via extension of scalars there is a natural isomorphism
S−1M≅(S−1R)⊗RM.
Examples
Localizing a quotient. If I⊆R is an ideal, then
S−1(R/I)≅(S−1R)/(S−1I),
where S−1I denotes the image of I in S−1R.
Torsion killed by localization. Take R=Z, M=Z/nZ, and localize at S=Z∖(p) (so S−1Z=Z(p)).
If p∤n, then n∈S becomes a unit, so S−1M=0.
If p∣n, then S−1M≅Z(p)/nZ(p), which is generally nonzero.
Making an element invertible forces a module to vanish. Let R=k[x], M=R/(x), and S={1,x,x2,…}. In S−1R the element x is a unit, but x annihilates M, so S−1M=0.