Definition 3.49. A category C consists of a class of objects (denoted by Obj C) and a set of morphisms HomC(A, B) for every pair of objects A, B of C such that
1. (Composition) there exists a function HomC(B, C) × HomC(A, B) → HomC(A, C) sending (f, g) 7→ f ◦ g for all objects A, B, C.
2. (Associativity) (f g)h = f (gh) for all morphisms f, g, h where (f g)h is defined.
3. (Identity) For all objects A of C there exists 1A∈ HomC(A, A) such that for all objects B of C we have 1Af = f for all f ∈ HomC(B, A) and f 1A= f for all f ∈ HomC(A, B).
Examples.
1. The category of sets: <<Sets>> has sets as objects and functions as morphisms.
2. The category of groups: <<Groups>> has groups as objects and group homomorphisms as morphisms. This category has the subcategory <<Abel>> of abelian groups. Note that a subcategory is called a full subcategory if it retains all of the morphisms.
3. For a commutative ring R, the category of R−algebras: <<R−algebra>> has R−algebras as objects and R−algebra homomorphisms (φ : S → T where S, T are R−algebras such that φ is a ring homomorphism where φ(rs) = rφ(s) for all r ∈ R) as the set of morphisms.
Note. Every ring is a Z−algebra. Thus <<Z−algebras>> = <<Rings>>.
4. For a commutative ring R, the category of left R−modules is written << R−mod>> and the category of right R−modules is written << mod−R >>.
Special Cases
(a) << Z−mod>> = << Abel>>
(b) If k is a field, << k−mod>> = << k−vector spaces>>
Definition 3.50. Let C and D by categories. A (covariant) functor F : C → D is a rule which associates to each object A of C an object F (A) of D and for each morphism f ∈ HomC(A, B) a morphism F (f ) ∈ HomD(F (A), F (B)) with the following properties:
1. F (f g) = F (f )F (g) for all morphisms f, g of C where f g is defined. 2. F (1A) = 1F (A) for all objects A of C.
Examples.
1. The forgetful functor F :<<Groups>>→<<Sets>> defined by sending a group G to the set G and the group homomorphism g to the function g. Another forgetful functor is F0:<< R−mod>>→<<Abel>> .
2. The Localization functor: F :<< R−mod>>→<< RS−mod>> where F (M ) = MS and F (f ) = f1.
3. The Modding Out functor: Let I be a 2-sided ideal of R. Then we can define F :<< R−mod>>→<< R/I−mod>> by F (M ) = M/IM and for an R−homomorphism f : M → M, F (f ) : M/IM → N/IN where m + IM 7→ f (m) + IN.
Note. You can mod out by a left ideal, however the functor would then be << R−mod>>→<< R−mod>> . Definition 3.51. Let M, N be left R−modules. Then HomR(M, N ) denotes the set of left R−module homomorphisms from M → N.
Remarks.
1. HomR(M, N ) is an abelian group.
2. Generally, HomR(M, N ) is not a left R−module, unless R is commutative.
3. Let M be a left R−module. Define a functor HomR(M, −) :<< R−mod >>→<< Abel >> by HomR(M, −)(N ) = HomR(M, N ) and if f : N1→ N2is an R−module homomorphism, then f∗:= HomR(M, −)(f ) : HomR(M, N1) →
HomR(M, N2) defined by g 7→ f g. Note that (f g)∗= f∗g∗and (1N)∗= 1HomR(M,N )(and thus it really is a functor).
Definition 3.52. A contravariant functor F : C → D is a rule which associates to each object A of C an object F (A) of D and for every pair of objects A, B of C a map HomC(A, B) → HomD(F (B), F (A)) defined by f 7→ F (f ) such that F (f g) = F (g)F (f ) and F (1A) = 1F (A).
Example. Let N be a left R−module. Define the contravariant functor HomR(−, N ) :<< R − mod >>→<< Abel >> by M 7→ HomR(M, N ) and (f : M1 → M2) 7→ (f∗ : HomR(M2, N ) → HomR(M1, N )) where g 7→ gf. One can check
that (f g)∗= g∗f∗.
Definition 3.53. Let F be a functor (of either variance) on module categories. We say F is additive if for every pair of objects A, B of the initial category, the map F : HomC(A, B) → HomD(F (A), F (B)) (or F : HomC(A, B) → HomD(F (B), F (A))) is a group homomorphism, that is, F (f + g) = F (f ) + F (g) for all f, g ∈ HomC(A, B).
Remarks.
1. Localization, Modding Out, and the Hom functors are all additive.
2. Suppose A−→ Bf −→ C is exact and let F be an additive covariant functor. Consider F (A)g −−−→ F (B)F (f ) −−−→ F (C).F (g) In general, this is not exact - but we do still get imF (f ) ⊆ kerF (g).
Proof. This is equivalent to showing F (g)F (f ) = 0. Of course, F (g)F (f ) = F (gf ) = F (0) = 0 as F is additive (F (0) = F (0) + F (0) implies F (0) = 0).
Definition 3.54. As additive functor on module categories is exact if whenever A −→ Bf −→ C is exact in the initialg category, then F (A)−−−→ F (B)F (f ) −−−→ F (C) is exact (or in the contravariant case F (C) → F (B) → F (A) is exact).F (g) Suppose F is covariant. Say F is left exact if
0 → A → B → C exact implies 0 → F (A) → F (B) → F (C) is exact and F is right exact if
A → B → C → 0 exact implies F (A) → F (B) → F (C) → 0 is exact. Suppose F is contravariant. Say F is left exact if
A → B → C → 0 exact implies 0 → F (C) → F (B) → F (A) is exact and F is right exact if
0 → A → B → C exact implies F (C) → F (B) → F (A) → 0 is exact. Proposition 3.55. Let F be an additive functor. TFAE
1. F is exact
2. F takes short exact sequences to short exact sequences 3. F is both left and right exact.
Remark. We’ve shown localization is an exact covariant functor.
Proposition 3.56. The modding out functor is right exact, but not generally exact.
Proof. Let I be a left ideal of R, L−→ Mf −→ N → 0 an exact sequence of R−modules. Consider L/ILg −→ M/IMf −→g N/IN → 0 where f (` + IL) = f (`) + IM and g(m + IM ) = g(m) + IN. As g is onto, so is g. Also, imf ⊆ ker g as modding out is an additive functor. So we need only show imf ⊇ ker g. Let x ∈ ker g. Then g(x) = g(x) = 0 which implies g(x) ∈ IN. Thus there exists ij∈ I, nj∈ N such that g(x) =
Pk
j=1ijnj. Let uj ∈ M such that g(uj) = nj. Then g(x) =Pujg(uj) = g( P ijuj). Thus g(x − P ijuj) = 0 which implies x − P
ijuj ∈ ker g = imf. Let ` ∈ L such that f (`) = x −Pijuj. Then f (`) = x ∈ imf .
To show it is not always left exact, consider 0 → Z−→ Z where n 7→ 2n. Modding out by (2) gives us 0 → Z/2Z2 −→ Z/2Z2 where n 7→ 2n = 0. Thus the map is not injective.
Proposition 3.57. Let M be a left R−module. Then HomR(M, −) and HomR(−, M ) are both left exact, but not generally exact.
Proof. We will prove only for HomR(M, −). Let 0 → A f
−→ B −→ C be exact and consider 0 → Homg R(M, A) f∗
−→ HomR(M, B)−→ Homg∗ R(M, C). As f is 1-1, we have f h = f∗(h) = 0 which implies h = 0. Thus f∗is 1-1. By additivity, imf∗⊆ ker g∗. Thus we need only show imf∗⊇ ker g∗. Let h ∈ ker g∗ where h : M → B. So g∗(h) = gh = 0. This says imh ⊆ kerg = imf. Thus for all m ∈ M there exists a unique am∈ A such that f (am) = h(m). Define k : M → A by k(m) = am. Then k ∈ HomR(M, A) and f∗(k) = h ∈ imf∗.
To show it is not always right exact, consider Z−→ Z → Z/2Z → 0. This gives us Hom2 Z(Z/2Z, Z) → HomZ(Z/2Z, Z) →
HomZ(Z/2Z, Z/2Z) → 0. Now, the first two modules are 0 and the last is isomorphic to Z/2Z. Thus it does not preserve
surjectivity.
Proposition 3.58. Let R be a ring and P a left R−module. Then P is projective if and only if HomR(P, −) is exact. Proof. We will only prove the forward direction. The backward direction is similar. Let 0 → A−→ Bf −→ C → 0 be exactg and apply the Hom functor:
0 → Hom(P, A) f∗
−→ Hom(P, B) g∗
−→ Hom(P, C) → 0.
By the previous proposition, it is enough to show g∗ is onto. Let h ∈ HomR(P, C). By the definition of projective, there exists k : P → B such that gk = h which implies g∗(k) = h. Thus h ∈ img∗and is thus onto.