Homological algebra (Math 239)

112 Pages·2015·0.552 MB·English

by  Robert Boltje

Lecture Notes
Math 239 (Algebra)
Spring 2015
(Robert Boltje, UCSC)
Contents
1 Categories and Functors 1
2 Simplicial and Semi-Simplicial Objects in a Category 16
3 Chain Complexes and Homology 23
4 Chain Complexes and Homotopy 33
5 The Long Exact Homology Sequence 43
6 The Mapping Cone 51
7 Extensions of Modules 59
8 Projective Resolutions 68
9 Derived Functors 75
10 The Functors Ext and Tor 82
11 Double Complexes and Tensor Products of Chain Complexes 92
12 Group Homology and Group Cohomology 102
1 Categories and Functors
1.1 Definition A category C consists of
• a class Ob(C), whose elements are called the objects of C,
• for any two objects C,C(cid:48) ∈ Ob(C), a set Hom (C,C(cid:48)), called the set of morphisms
C
from C to C(cid:48), and,
• for any three objects C,C(cid:48),C(cid:48)(cid:48) ∈ Ob(C), a composition map
Hom (C,C(cid:48))×Hom (C(cid:48),C(cid:48)(cid:48)) → Hom (C,C(cid:48)(cid:48)), (f,g) (cid:55)→ g ◦f .
C C C
These data are subject to the following axioms:
(C1) If (C ,C(cid:48)) and (C ,C(cid:48)) are different pairs of objects of C then Hom (C ,C(cid:48)) and
1 1 2 2 C 1 1
Hom (C ,C(cid:48)) are disjoint sets. In other words, each morphism has a uniquely
C 2 2
determined domain and a uniquely determined codomain.
(C2) Composition of morphisms is associative.
(C3) ForeachobjectC ofCthereexistsamorphism1 ∈ Hom (C,C)suchthat1 ◦f =
C C C
f, for any A ∈ Ob(C) and any f ∈ Hom (A,C), and g ◦1 = g for any A ∈ Ob(C)
C C
and any morphism g ∈ Hom (C,A)).
C
We will often write f: C → C(cid:48) to indicate that f ∈ Hom (C,C(cid:48)).
C
1.2 Examples (a) Sets and functions between sets form a category, denoted by Set.
There is also a category of finite sets, denoted by set.
(b) There are categories Gr, gr, Ri, Ri , Ab of groups, finite groups, rings, commutative
c
rings, and abelian groups, together with their respective homomorphisms. We assume
that rings are always associative and have a multiplicative identity element, and that ring
homomorphisms preserve identity elements.
(c) For every ring R, one has a category Mod, the category of left R-modules.
R
Similarly, Mod denotes the category of right R-modules. If also S is a ring, we de-
R
note by Mod the category of (R,S)-bimodules. An (R,S)-bimodule is an abelian
R S
group M which has a left R-module structure and a right S-module structure such that
(rm)s = r(ms) for all r ∈ R, s ∈ S, and m ∈ M. Homomorphisms between (R,S)-
bimodules are group homomorphism which are both left R-module homomorphisms and
right S-module homomorphisms.
1
(d) Top denotes the category of topological spaces and continuous maps.
(e) If C and D are categories then their product category C×D is defined as follows:
Its objects are pairs (C,D), where C ∈ Ob(C) and D ∈ Ob(D), and the set of morphisms
between two objects (C,D) and (C(cid:48),D(cid:48)) consists of all pairs (f,g) with f ∈ Hom (C,C(cid:48))
C
and g ∈ Hom (D,D(cid:48)). Composition is defined componentwise.
D
1.3 Definition Let C and D be categories.
(a) A covariant functor F: C → D consists of functions of the following form:
• A function Ob(C) → Ob(D), denoted by C (cid:55)→ F(C), and,
• for any two objects C,C(cid:48) of C, a function Hom (C,C(cid:48)) → Hom (F(C),F(C(cid:48))),
C D
again denoted by f (cid:55)→ F(f).
These functions are subject to the following axioms:
(F 1) F(g◦f) = F(g)◦F(f),forallC,C(cid:48),C(cid:48)(cid:48) ∈ Ob(C)andallf: C → C(cid:48) andg: C(cid:48) → C(cid:48)(cid:48)
∗
in C.
(F 2) F(1 ) = 1 for all C ∈ Ob(C).
∗ C F(C)
(b) Acontravariant functor F: C → D consists of functions of the following form:
• A function Ob(C) → Ob(D), denoted by C (cid:55)→ F(C), and,
• for any two objects C,C(cid:48) of C, a function Hom (C,C(cid:48)) → Hom (F(C(cid:48)),F(C)),
C D
again denoted by f (cid:55)→ F(f).
These functions are subject to the following axioms:
(F∗1) F(g◦f) = F(f)◦F(g),forallC,C(cid:48),C(cid:48)(cid:48) ∈ Ob(C)andallf: C → C(cid:48) andg: C(cid:48) → C(cid:48)(cid:48)
in C.
(F∗2) F(1 ) = 1 for all C ∈ Ob(C).
C F(C)
We make the convention that the word ”functor” by itself always means ”covariant
functor”.
1.4 Examples (a) For any category C one has an obvious identity functor Id : C → C.
C
(b) If C, D, and E are categories and F: C → D and G: D → E are covariant or
contravariant functors then one can define G ◦ F: C → E on objects by C (cid:55)→ G(F(C))
and on morphisms by f (cid:55)→ G(F(f)). If both F and G are of the same type (covariant
2
or contravariant) then G ◦ F is covariant. If they are of different type then G ◦ F is
contravariant.
(c) The unit group functor U: Ri → Gr maps a ring R to its unit group U(R), and a
ring homomorphism f: R → S to a the group homomorphism f: U(R) → U(S). More
generally, for every positive integer n, one has the functor GL : Ri → Gr which maps a
n
ring R to the general linear group GL (R) and a ring homomorphism f: R → S to the
n
group homomorphism GL (f): GL (R) → GL (S), (r ) (cid:55)→ (f(r )).
n n n ij ij
(d) Let R be a ring and let M be in Mod. There is a covariant functor
R
Hom (M,−): Mod → Ab
R R
N (cid:55)→ Hom (M,N)
R
(cid:16) (cid:17)
(f: N → N(cid:48)) (cid:55)→ Hom (M,f) = f : Hom (M,N) → Hom (M,N(cid:48))
R ∗ R R
g (cid:55)→ f ◦g
Here Hom (M,N) is an abelian group through (f +f(cid:48))(m) := f(m)+f(cid:48)(m) for f,f(cid:48) ∈
R
Hom (M,N) and m ∈ M.
R
Similarly, for N ∈ Mod, one has a contravariant functor
R
Hom (−,N): Mod → Ab
R R
M (cid:55)→ Hom (M,N)
R
(cid:16) (cid:17)
(f: M → M(cid:48)) (cid:55)→ Hom (f,N) = f∗: Hom (M(cid:48),N) → Hom (M,N)
R R R
g (cid:55)→ g ◦f
If R is commutative, each of these functors can be considered as functor from Mod
R
to Mod. In fact, in this case Hom (M,N) is an R-module via (rf)(m) := rf(m) for
R R
f ∈ Hom (M,N), r ∈ R and m ∈ M, and Hom (M,f) and Hom (f,N) are R-module
R R R
homomorphisms.
More generally, if also S is a ring and M is an (R,S)−bimodule then Hom (M,−)
R
can be viewed as a functor form Mod to Mod, where Hom (M,N) is viewed as a left
R S R
S-module via (sf)(m) := f(ms), for f ∈ Hom (M,N), s ∈ S and m ∈ M. Similarly,
R
if N is an (R,S)-bimodule then Hom (−,N) can be considered as a functor from Mod
R R
to Mod , where Hom (M,N) is considered as right S-module via (fs)(m) := f(m)s for
S R
f ∈ Hom (M,N), s ∈ S and m ∈ M.
R
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(e) Specializing N = R in (d), we obtain a contravariant functor
Hom (−,R): Mod → Mod
R R R
M (cid:55)→ Hom (M,R)
R
(cid:16) (cid:17)
(f: M → M(cid:48)) (cid:55)→ Hom (f,R) = f∗: Hom (M(cid:48),R) → Hom (M,R)
R R R
g (cid:55)→ g ◦f
This functor is often called the R-dual functor.
If R is a k-algebra for a commutative ring k, there is a different type of functor, called
the k-dual functor:
Hom (−,k): Mod → Mod
k R R
M (cid:55)→ Hom (M,k)
k
(cid:16) (cid:17)
(f: M → M(cid:48)) (cid:55)→ Hom (f,k) = f∗: Hom (M(cid:48),k) → Hom (M,k)
k k k
g (cid:55)→ g ◦f
Here Hom (M,k) is a right R-module via (fr)(m) = f(rm) for f ∈ Hom (M,k), m ∈ M
k k
and r ∈ R. For certain types of algebras, so-called symmetric algebras R, the R-dual
and the k-dual functors are isomorphic in a precise sense which will be defined later, see
1.24(b). Group algebras for instance are symmetric algebras.
(f) If R is a ring and M ∈ Mod we obtain a covariant functor
R
−⊗ M: Mod → Ab
R R
L (cid:55)→ L⊗ M
R
(f: L → L(cid:48)) (cid:55)→ (f⊗ : L⊗ M → L(cid:48) ⊗ M)
idM R R
Simiarly, for M ∈ Mod , one obtains a covariant functor
R
M ⊗ −: Mod → Ab.
R R
If M ∈ Mod is an (R,S)-bimodule for an additional ring S, then one obtains functors
R S
−⊗ M: Mod → Mod and M ⊗ −: Mod → Mod.
R R S S S R
1.5 Definition Let C be a category, let B,C ∈ Ob(C), and let g ∈ Hom (B,C).
C
(a) The morphism g is called a monomorphism if g◦f = g◦f implies f = f for all
1 1 1 2
A ∈ Ob(C) and f ,f ∈ Hom (A,B).
1 2 C
4
(b) The morphism g is a called an epimorphism if h ◦g = h ◦g implies h = h for
1 2 1 2
all D ∈ Ob(C) and all h ,h ∈ Hom (C,D).
1 2 C
(c) g is called an isomorphism, if there exists a morphism f: C → B in C such that
f ◦g = 1 and g ◦f = 1 . In this case, f is uniquely determined by g and denoted by
B C
g−1.
∼
(d) The objects B and C of C are called isomorphic (notation B = C), if there exists
an isomorphism f: B → C in C.
1.6 Remark (a) In the category Set, a morphism is a monomorphism (resp. epimor-
phism, resp. isomorphism) if and only if it is injective (resp. surjective, resp. bijective).
The same holds for instance in the categories Gr and Mod; but not in Ri.
R
(b) Every isomorphism is a monomorphism and an epimorphism. But the converse is
not true: Consider the inclusion Z → Q in Ri.
(c) If F: C → D is a covariant or contravariant functor and f: C → C(cid:48) is an isomor-
phism in C, then F(f) is an isomorphism in D. However, the class of monomorphisms or
epimorphisms is not preserved under covariant or contravariant functors.
(d) Every category C has an opposite category C◦. Its objects are the same as those
in C. But when viewed as objects in C◦ we denote them by C◦. For two objects A◦
and B◦ of C◦, one sets Hom (A◦,B◦) := Hom (B,A). We denote f ∈ Hom (B,A) by
C◦ C C
f◦: A◦ → B◦, if it is viewed as a morphism in C◦. Composition in C◦ is defined as follows:
For A,B,C ∈ Ob(C) and f◦: A◦ → B◦ and g◦: B◦ → C◦, one has g◦ ◦f◦ := (f ◦g)◦.
Note that a morphism f◦: A◦ → B◦ in C◦ is a monomorphim in C◦ if and only if
f: B → A is an epimorphism in C. Morover, f◦ is an epimorphism in C◦ if and only if f
is amonomorphisminC. We saythat theconcepts of‘monomorphism’and ‘epimorphism’
are dual to each other: The one translates into the other in the opposite category.
There is an obvious contravariant functor C → C◦, sending an object A of C to the
object A◦ of C◦ and a morphism f in C to f◦. Moreover, there is a contravariant functor
C◦ → C defined by A◦ (cid:55)→ A and f◦ (cid:55)→ f. The compositions of these two functors yield
the identity functors on C and on C◦. For any category D, there is a bijection between
the collection of contravariant functors C → D and the collection of covariant functors
C◦ → D, given by precomposing with the above functors. This way one can always switch
between contravariant and covariant functors if necessary. For instance, for any ring R,
one can consider Hom (−,−) as a covariant functor
R
Hom (−,−): Mod◦ × Mod → Ab.
R R R
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1.7 Definition LetC beacategory. AnobjectAofCiscalledaninitial object(resp.final
object) of C if |Hom (A,B)| = 1 (resp. |Hom (B,A)| = 1) for all B ∈ Ob(C). If A is both
C C
an initial and a final object in C then we call A a zero object.
1.8 Remark (a)IfAandA(cid:48) aretwoinitial(resp.final)objectsofacategoryCthenthere
exist unique morphisms in Hom (A,A(cid:48)) and in Hom (A(cid:48),A). These two maps are inverse
C C
isomorphisms, since Hom (A,A) and Hom (A(cid:48),A(cid:48)) contain only the respective identity
C C
morphism. Thus, initial (resp. final) objects in C are unique up to unique isomorphism,
and we may speak of the initial (resp. final) object in C (if it exists). This also implies
that zero objects are unique up to unique isomorphism. The zero object, if it exists, is
usually denoted by 0.
(b) In the category Set, the empty set is an initial object and any set with one element
is a final object. In the category Gr, the trivial group is a zero object. Similarly, the zero
module is a zero object in the category Mod, for every ring R.
R
1.9 Definition Let C be a category and let A , i ∈ I, be objects of C. A product
i
(P,(π ) ) of A , i ∈ I, consists of an object P of C together with morphisms π : P → A ,
i i∈I i i i
i ∈ I, inC, calledprojections, suchthatforanyobjectQofCandanyfamilyofmorphisms
f : Q → A , i ∈ I, in C, there exists a unique morphism f: Q → P with the property
i i
π ◦f = f for all i ∈ I. Note that if (P,(π ) ) is a product of A , i ∈ I, and Q ∈ Ob(C),
i i i i∈I i
then
Hom (Q,P) → × Hom (Q,A ), f (cid:55)→ (π ◦f) ,
C C i i i∈I
i∈I
is a bijection. In particular, for any two morphisms f,g: Q → P in C one has f = g if
and only if π ◦f = π ◦g for all i ∈ I.
i i
1.10 Proposition Let C be a category and let A , i ∈ I, be a family of objects in C.
i
Assume that (P,(π ) ) and (P(cid:48),(π(cid:48)) ) are products of A , i ∈ I. Then there exists a
i i∈I i i∈I i
unique morphism f: P → P(cid:48) in C such that π(cid:48) ◦ f = π for all i ∈ I. Moreover, f is
i i
an isomorphism. Thus, products are unique up to unique isomorphism (if they exist).
Therefore, we denote any product of A , i ∈ I, by (cid:81) A (without using a notation for
i i∈I i
the projections).
Proof Since (P(cid:48),(π(cid:48)) ) is a product of A , i ∈ I, there exists a unique f: P → P(cid:48)
i i∈I i
in C such that π(cid:48) ◦ f = π for all i ∈ I. Similarly, since (P,(π ) ) is a product of A ,
i i i i∈I i
i ∈ I, there exists a unique morphism g: P(cid:48) → P in C such that π ◦g = π(cid:48) for all i ∈ I.
i i
Therefore, π ◦ (g ◦ f) = (π ◦ g) ◦ f = π(cid:48) ◦ f = π = π ◦ 1 for all i ∈ I. This implies
i i i i i P
g ◦f = 1 . Similarly, one shows that f ◦g = 1 .
P P(cid:48)
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1.11 Definition Let C be a category and let A , i ∈ I, be a family of objects of C. A
i
coproduct (C,(ι ) ) of A , i ∈ I, in C consists of an object C of C together with a family
i i∈I i
of morphisms ι : A → C in C, called the injections, satisfying the following property. For
i i
any object D in C and any family of morphisms f : A → D, i ∈ I, in C, there exists a
i i
unique morphism f: C → D with f ◦ι = f , for all i ∈ I.
i i
1.12 Remark Assume the situation of the previous definition. Note that (C,(ι ) ) is
i i∈I
a coproduct of A , i ∈ I, in C, if and only if (C◦,(ι◦) ) is a product of A◦, i ∈ I, in
i i∈I i
the opposite category C◦. Thus, coproducts and products are dual concepts. Now Propo-
sition 1.10 applied to C◦ implies that coproducts are unique up to unique isomorphism.
We denote the coproduct of A , i ∈ I, by (cid:96) A (without including a notation for the
i i∈I i
injections).
1.13 Examples (a) In Set, the cartesian product × A of sets A , i ∈ I, together
i∈I i i
with the usual projection maps π : × A , is a categorical product in the sense of
i i∈I i
Definition 1.9. The disjoint union (cid:85) A together with the inclusions ι : A → (cid:85)A ,
i∈I i i i i
i ∈ I, is a coproduct.
(b)InthecategoryGr,thecartesianproduct× A ,togetherwiththeusualprojection
i∈I i
maps, is again a categorical product. Coproducts also exist but are more complicated to
define.
(c) Let R be a ring and let M , i ∈ I, be objects in Mod. The product of M , i ∈ I,
i R i
is given by × M and the usual projection maps. The coproduct of M , i ∈ I, is the
i∈I i i
submodule of × M consisting of those elements that have only finitely many non-zero
i∈I i
components. The injections are the usual embeddings. It is instructive to check why the
cartesian product of M together with the usual inclusion maps is not a coproduct. Note
i
that the images of the inclusion maps M → × M do not generate × M . If I is finite
j i∈I i i∈I i
then the underlying modules of the product and coproduct coincide.
1.14 Definition (a) An additive category is a category C together with abelian group
structures on each morphism set Hom (A,B), A,B ∈ Ob(C) satisfying the following
C
axioms:
(Add 1) C has a zero object.
(Add 2) C has finite products and coproducts.
(Add 3) The composition of morphisms is biadditive.
7
(b) Let C and D be additive categories. An additive functor from C to D is a covariant
functor F: C → D with the property that for any two objects A and B of C, the map
F: Hom (A,B) → Hom (F(A),F(B)), f (cid:55)→ F(f), is a group homomorphism. Similarly,
C D
one defines contravariant additive functors.
1.15 Remark (a) If C is an additive category then also C◦ is an additive category. Here,
the group structure on Hom (A◦,B◦) is the same as the one on Hom (B,A).
C◦ C
(b) For a commutative ring k, there is also the notion of a k-linear category C. The
morphism sets Hom (A,B) have the additional structure of a k-module and composition
C
is k-bilinear. Moreover, a functor between k-linear categories is said to be k-linear if it is
k-linear on morphism sets. The notions of an additive category and additive functors are
the special case were k = Z.
(c) If C is an additive category then also C◦ is an additive category. The group
structure on morphism sets of C◦ is defined by f◦ +g◦ := (f +g)◦ for A,B ∈ Ob(C) and
f,g ∈ Hom (A,B).
C
1.16 Examples (a) The category Set can’t be made into an additive category, since it
has no zero object.
(b) The category Ab is additive. More generally, Mod is additive for an ring R. If R
R
is a k-algebra over a commutative ring k then Mod is a k-linear category.
R
1.17 Proposition Assume that C is an additive category and that A , i ∈ I, is a finite
i
family of objects in C.
(a) If (P,(π ) ) is a product of A , i ∈ I, in C then there exist unique morphisms
i i∈I i
ι : A → P, i ∈ I, in C such that
i i

0 if i (cid:54)= j,
π ◦ι = (1.17.a)
i j
1Ai if i = j,
for i,j ∈ I. Moreover,
(cid:88)
ι ◦π = 1 (1.17.b)
i i P
i∈I
and (P,(ι ) ) is a coproduct of A , i ∈ I, in C.
i i∈I i
(b) If (C,(ι ) ) is a coproduct of A , i ∈ I, in C then there exist unique morphisms
i i∈I i
π : C → A , i ∈ I, in C such that Equation (1.17.a) holds. Moreover, Equation (1.17.b)
i i
holds and (C,(π ) ) is a product of A , i ∈ I, in C.
i i∈I i
8