7.5 The Adjointness Relationship

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Although we have not formally developed any category theory in these notes,

we shall, in this section, use some of the elementary language. Let R be a

ring and let RMod, Ab denote the categories of left R-modules and abelian

groups, respectively. Thus, if M is a fixed right R-module, then we have a

functor

M R − :R Mod −! Ab.

In an entirely similar way, for any fixed left R-module N, there is a functor

− R N : ModR ! Ab, where ModR is the category of right R-modules.

Next we consider a functor Ab !R Mod, alluded to in Section 8 of Chapter

5. Indeed, if M is a fixed right R-module, we may define

HomZ(M,−) : Ab !R Mod.

Indeed, note that if A is an abelian group, then by Exercise 10 of Section 5.8,

HomZ(M,A) is a left R-module via (r · f)(m) = f(mr). For the fixed right

R-module M, the functors M R − and HomZ(M,−) satisfy the following

important adjointness relationship.

Theorem 7.5.1 (Adjointness Relationship) If M is a right R-module,

N is a left R-module, and if A is an abelian group, there is a natural equivalence

of sets:

HomZM R N,A) _=Set HomR(N,HomZ(M,A)).

Indeed, in the above, the relevant mappings are as follows:

f 7! (n 7! (m 7! f(m  n))), g 7! (m  n 7! g(n)(m)),

where f 2 HomZ(M R N,A), g 2 HomR(N,HomZ(M,A)).

In general if C,D are categories, and if F : C ! D, G : D ! C are

functors, we say that F is left adjoint to G (and that G is right adjoint to

F) if there is a natural equivalence of sets

HomD(F(X), Y ) _=Set HomC(X, F(Y )),

where X is an object of C and Y is an object of D. Thus, we see that the

functor M R − is left adjoint to the functor HomZ(M,−).

One of the more important consequences of the above is in Exercise 1

below.

7.5. THE ADJOINTNESS RELATIONSHIP 173

Exercises 7.5

1. Using the above adjointness relationship, interpret and prove the following:

M R − preserves epimorphisms, and HomZ(M,−) preserves

monomorphisms.

2. Let C be a category and let μ : A ! B be a morphism. We say that

μ is a monomorphism if whenever A0 is an object with morphisms

f : A0 ! A, g : A0 ! A such that μ _ f = μ _ g : A0 ! B, then

f = g : A0 ! A. In other words, monomorphisms are those morphisms

that have “left inverses.” Similarly, epimorphisms are those morphisms

that have right inverses. Now assume that C,D are categories, and that

F : C ! D, G : D ! C are functors, with F left adjoint to G. Prove

that F preserves epimorphisms and that G preserves monomorphisms.

3. Let i : Z ,! Q be the inclusion homomorphism. Prove that in the

category of rings, i is an epimorphism. Thus an epimorphism need

not be surjective.

4. Let V,W be F-vector spaces and let V _ be the F-dual of V . Prove

that there is a vector space isomorphism V _ F W _= HomF(V,W).

5. Let G be a group. Exactly as in Section 5.10, we may define the integral

group ring ZG; (these are Z-linear combinations of group elements in

G). correspondingly, given a ring R we may form its group of units

U(R). Thus we have functors

Z : Groups −! Rings, U : Rings −! Groups.

Prove that Z is left adjoint to U.

6. Below are some further examples of adjoint functors. In each case you

are to prove that F is left adjoint to G.

(a)

Groups F −! Abelian Groups G

,! Groups;

F is the commutator quotient map.

174 CHAPTER 7. TENSOR PRODUCTS

(b)

Sets F −! Groups G −! Sets,

where F(X) = free group on X and G(H) is the underlying set

of the group H.

(c)

Integral Domains F −! Fields

G

,! Integral Domains;

F(D) is the field of fractions of D. (Note: for this example we

consider the morphisms of the category Integral Domains to

be restricted only to injective homomorphisms.)

(d)

K − V ector Spaces F −! K − Algebras G −! K − V ector Spaces;

F(V ) = T(V ), the tensor algebra of V and G(A) is simply the

underlying vector space structure of A.

(e)

Abelian Groups F −! Torsion Free Abelian Groups G

,! Abelian Groups;

F(A) = A/T(A), where T(A) is the torsion subgroup of A.

(f)

Left R − modules F −! Abelian Groups G −! Left R − modules;

F is the forgetful functor, G = HomZ(R,−).

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