INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ˇCADEK
6. More homological algebra
In this section we will deal with algebraic constructions leading to the definitions
of homology and cohomology groups with coefficients given in the previous sections.
At the end we use introduced notions to state and prove so called universal coefficient
theorems for singular homology and cohomology groups.
6.1. Functors and cofunctors. Let A and B be two categories. A functor t : A → B
assigns to every object x in A an object t(x) in B and to every morphism f : x → y in
A a morphism t(f) : t(x) → t(y) such that t(idx) = idt(x) and t(fg) = t(f)t(g).
A contravariant functor or briefly cofunctor t : A → B assigns to every object x in A
an object t(x) in B and to every morphism f : x → y in A a morphism t(f) : t(y) → t(x)
in B such that t(idx) = idt(x) and t(fg) = t(g)t(f).
Let R be a commutative ring with a unit element. The category of R-modules
and their homomorphisms will be denoted by R-Mod. R-GMod will be used for the
category of graded R-modules, R-Ch and R-CoCh will stand for the categories of chain
complexes and the category of cochain complexes of R-modules, respectively. For
R = Z the previous categories are Abelian groups Ab, graded Abelian groups GAb,
chain complexes of Abelian groups Ch and cochain complexes of Abelian groups CoCh,
respectively.
Homology H is a functor from the category R-Ch to the category R-GMod. Let t
be a functor from R-Mod to R-Mod which induces a functor t : Ch → Ch, and let s be
a cofunctor from R-Mod to R-Mod, which induces a cofunctor from Ch to CoCh. The
aim of this section is to say something about the functor H ◦t and the cofunctor H ◦s.
Model examples of such functors will be t(−) = − ⊗R M and s(−) = HomR(−, M)
for a fixed R-module M. We have already used these functors when we have defined
homology and cohomology groups with coefficients.
6.2. Tensor product. The tensor product A ⊗R B of two R-modules A and B is the
quotient of the free R-module over A × B and the ideal generated by the elements of
the form
r(a, b)−(ra, b), r(a, b)−(a, rb), (a1 +a2, b)−(a1, b)−(a2, b), (a, b1 +b2)−(a, b1)−(a, b2)
where a, a1, a2 ∈ A, b, b1, b2 ∈ B, r ∈ R. The class of equivalence of the element (a, b)
in A ⊗R B is denoted by a ⊗ b. The map ϕ : A × B → A ⊗R B, ϕ(a, b) = a ⊗ b is
bilinear and has the following universal property:
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Whenever an R-module C and a bilinear map ψ : A × B → C are given, there is
just one R-modul homomorphism Ψ : A ⊗R B → C such that the diagram
A × B
ψ
//
ϕ

C
A ⊗R B
Ψ
;;vvvvvvvvv
commutes. This property determines the tensor product uniquely up to isomorphism.
If f : A → C and g : B → D are homomorphisms of R-modules then (a, b) →
f(a) ⊗ g(b) is a bilinear map and the universal property above ensures the existence
and uniqueness of an R-homomorphism f ⊗ g : A ⊗R B → C ⊗R D with the property
(f ⊗ g)(a ⊗ b) = f(a) ⊗ g(b).
Homomorphisms between R-modules form an R-module denoted by HomR(A, B).
If R = Z, we will denote the tensor product of Abelian groups A and B without the
subindex Z, i.e. A ⊗ B, and similarly, the group of homomorfisms from A to B will
be denoted by Hom(A, B).
Exercise. Prove from the definition that
Z ⊗ Z = Z, Z ⊗ Zn = Zn, Zn ⊗ Zm = Zd(n,m), Zn ⊗ Q = 0
where d(m, n) is the greatest common divisor of n and m. Further compute
Hom(Z, Z), Hom(Z, Zn), Hom(Zn, Z), Hom(Zn, Zm).
6.3. Additive functors and cofunctors. A functor (or a cofunctor) t : Mod → Mod
is called additive if
t(α + β) = t(α) + t(β)
for all α, β ∈ HomR(A, B). Additive functors and cofunctors have the following prop-
erties.
(1) t(0) = 0 for any zero homomorphism.
(2) t(A ⊕ B) = t(A) ⊕ t(B)
(3) Every additive functor (cofunctor) converts short exact sequences which split
into short exact sequences which again split.
(4) Every additive functor (cofunctor) can be extended to a functor Ch → Ch
(cofunctor Ch → CoCh) which preserves chain homotopies (converts chain homotopies
to cochain homotopies).
Proof of (2) and (3). Consider a short exact sequence
0 → A
i
−→ B
j
−→ C → 0
which splits, i. e. there are homomorphisms p : B → A, q : C → B such that pi = idA,
jq = idC, ip + qj = idB. See 3.1. Applying an additive functor t we get a splitting
short exact sequence described by homomorphisms t(i), t(j), t(p), t(q).
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6.4. Exact functors and cofunctors. An additive functor (or an additive cofunctor)
t : Mod → Mod is called exact if it preserves short exact sequences.
Example. The functor t(−) = − ⊗ Z2 and the cofunctor s(−) = Hom(−, Z) are
additive but not exact. To show it apply them on the short exact sequence
0 → Z
2×
−→ Z → Z2 → 0.
On the other hand, the functor t(−) = − ⊗ Q from Ab to Ab is exact.
Lemma. Let (C, ∂) be a chain complex and let t : Mod → Mod be an exact functor.
Then
Hn(tC, t∂) = tHn(C, ∂).
Consequently, t converts all exact sequences into exact sequences.
Proof. Since t preserves short exact sequences, it preserves kernels, images and factors.
So we get
Hn(tC) =
Ker t∂n
Im t∂n+1
=
t(Ker ∂n)
t(Im ∂n+1)
= t
Ker ∂n
Im ∂n+1
= t(Hn(C)).
6.5. Right exact functors. An additive functor t : Mod → Mod is called right exact
if it converts any exact sequence
A
i
−→ B
j
−→ C → 0
into an exact sequence
tA
t(i)
−−→ tB
t(j)
−−→ tC → 0.
Theorem. Consider an R-module M. The functor t(−) = − ⊗R M from Mod to Ab
is right exact.
Proof. The exact sequence A
i
−→ B
j
−→ C → 0 is converted into the sequence
A ⊗R M
i⊗idM
−−−→ B ⊗R M
j⊗idM
−−−−→ C ⊗R M → 0.
It is clear that j ⊗idM is an epimorphism. According to the lemma below Ker(j ⊗idM )
is generated by elements b ⊗ m where b ∈ Ker j = Im i. Hence, Ker(j ⊗ idM ) =
Im(i ⊗ idM ).
Lemma. If α : A → A and β : B → B are epimorphisms, then Ker(α ⊗ β) is
generated by elements a ⊗ b where a ∈ Ker α or b ∈ Ker β.
For the proof see [Spanier], Chapter 5, Lemma 1.5.
6.6. Left exact cofunctors. An additive cofunctor t : Mod → Mod is called left
exact if it converts any exact sequence
A
i
−→ B
j
−→ C → 0
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into an exact sequence
O → tC
t(j)
−−→ tB
t(i)
−−→ tA.
Theorem. Consider an R-module M. The cofunctor t(−) = HomR(−, M) from Mod
to Mod is left exact.
The proof is not difficult and is left as an exercise.
6.7. Projective modules. An R-modul is called projective if for any epimorphism
p : A → B and any homomorphism f : P → B there is F : P → A such that the
diagram
A
p

P
F
??~~~~~~~~
f
// B

0
commutes. Every free R-module is projective.
6.8. Projective resolution. A projective resolution of an R-module A is a chain
complex (P, ε), Pi = 0 for i < 0 and a homomorphism α : P0 → A such that the
sequence
→ Pi
εi
−→ Pi−1 → · · · → P1
ε1
−→ P0
α
−→ A → 0
is exact. It means that
Hi(P, ε) =
0 for i = 0,
P0/ Im ε1 = P0/ Ker α ∼= A for i = 0.
If all Pi are free modules, the resolution is called free.
Lemma A. To every module there is a free resolution.
Proof. For module A denote F(A) a free module over A and π : F(A) → A a canonical
projection. Then the free resolution of A is constructed in the following way
P2 = F Ker ε1
//
ε2
--
Ker ε1
// P1 = F(Ker π) //
ε1
,,
Ker π // P0 = F(A) π
// A
Lemma B. Every Abelian group A has the projective resolution
0 → Ker π → F(A)
π
−→ A → 0.
Proof. Ker π as a subgroup of free Abelian group F(A) is free.
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Theorem. Consider a homomorphism of R-modules ϕ : A → A . Let (Pn, εn) and
(Pn, εn) be projective resolutions of A and A , respectively. Then there is a chain
homomorphism ϕn : (P, ε) → (P , ε ) such that the diagram
. . .
ε3
// P2
ε2
//
ϕ2

P1
ε1
//
ϕ1

P0
α //
ϕ0

A //
ϕ

0
. . .
ε3
// P2
ε2
// P1
ε1
// P0
α // A // 0
commutes. Moreover, any two such chain homomorphism (P, ε) → (P , ε ) are chain
homotopic.
Proof of the first part. α is an epimorphism and P0 is projective. Hence there is ϕ0 :
P0 → P0 such that the first square on the right side commutes.
Since α (ϕ0ε1) = ϕ(αε1) = ϕ ◦ 0 = 0, we get that
Im(ϕ0ε) ⊆ Ker α = Im ε1.
ε1 : P1 → Im ε1 is an epimorhism and P1 is projective. Hence there is ϕ1 : P1 → P1
such that the second square in the diagram commutes.
The proof of the rest of the first part proceeds in the same way by induction.
Proof of the second part. Let ϕ∗ and ϕ∗ be two chain homomorphisms making the
diagram above commutative. Since α (ϕ0 − ϕ0) = α(ϕ − ϕ) = 0, we have
Im(ϕ0 − ϕ0) ⊆ Ker α = Im ε1.
Therefore there exists s0 : P0 → P1 such that ε1s0 = ϕ0 − ϕ0.
Next, ε1(ϕ1 − ϕ1 − s0ε1) = ε1(ϕ1 − ϕ1) − ε1s0ε1 = (ϕ0 − ϕ0)ε1 − (ϕ0 − ϕ0)ε1 = 0,
hence
Im(ϕ0 − ϕ0 − s0ε1) ⊆ Ker ε1 = Im ε2,
and consequently, there is s1 : P1 → P2 such that
ε2s1 = ϕ1 − ϕ1 − s0ε1.
The rest proceeds by induction in the same way.
6.9. Derived functors. Consider a right exact functor t : Mod → Mod and a homomorphism
of R-modules ϕ : A → A . Let (P, ε) and (P , ε ) be projective resolutions
of A and A , respectively, and let ϕ∗ : (P, ε) → (P , ε ) be a chain homomorphism
induced by ϕ. The derived functors ti : Mod → Mod of the functor t are defined
tiA = Hi(tP, tε)
tiϕ = Hi(tϕ).
The functor t0 is equal to t since
t0A = tP0/ Im tε1 = tP0/ Ker tα ∼= tA.
Using the previous theorem we can easily show that the definition does not depend on
the choice of projective resolutions and a chain homomorphism ϕ∗.
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Definition. The i-th derived functors of the functor t(−) = − ⊗R M is denoted
TorR
i (−, M).
If R = Z, the index Z in the notation will be omitted.
Example. Let R = Z. Any Abelian group A has a free resolution with Pi = 0 for
i ≥ 2. Hence
Tori(A, B) = 0 for i ≥ 2.
Hence we will omit the index 1 in Tor1(A, B). We have
(1) Tor(A, B) = 0 for any free Abelian group A.
(2) Tor(A, B) = 0 for any free Abelian group B.
(3) Tor(Zm, Zn) = Zd(m,n) where d(m, n) is the greatest common divisor of m and
n.
(4) Tor(−, B) is an additive functor.
The proof based on the definition is not difficult and is left to the reader as an exercise.
6.10. Derived cofunctors. Consider a left exact cofunctor t : Mod → Mod and a
homomorphism of R-modules ϕ : A → A . Let (P, ε) and (P , ε ) be projective resolutions
of A and A , respectively, and let ϕ∗ : (P, ε) → (P , ε ) be a chain homomorphism
induced by ϕ. The derived cofunctors ti
: Mod → Mod of the functor t are defined
ti
A = Hi
(tP, tε)
ti
ϕ = Hi
(tϕ).
The functor t0
is equal to t since
t0A = Ker tε1 = Im tα = tA.
Using Theorem 6.8 we can easily show that the definition does not depend on the
choice of projective resolutions and a chain homomorphism ϕ∗.
Definition. The i-th derived functors of the functor t(−) = HomR(−, M) is denoted
Exti
R(−, M).
If R = Z, the index Z in the notation will be omitted.
Example. Let R = Z. Since every Abelian group A has a free resolution with Pi = 0
for i ≥ 2,
Exti
(A, B) = 0 for i ≥ 2.
Hence we will write Ext(A, B) for Ext1
(A, B). We have
(1) Ext(A, B) = 0 for any free Abelian group A.
(2) Ext(Zn, Z) = Zn.
(3) Ext(Zm, Zn) = Zd(m,n) where d(m, n) is the greatest common divisor of m and
n.
(4) Ext(−, B) is an additive cofunctor.
The proof is the application of the definition and it is again left to the reader.
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6.11. Universal coefficient theorems. In this paragraph we first express the cohomology
groups Hn
(X; G) with the aid of functors Hom and Ext using the homology
groups H∗(X).
Theorem A. If a free chain complex C of Abelian groups has homology groups Hn(C),
then the cohomology groups Hn
(C; G) of the cochain complex Cn
= Hom(Cn, G) are
determined by the following split short exact sequence
0 → Ext(Hn−1(C), G) → Hn
(C; G)
h
−→ Hom(Hn(C), G) → 0
where h[f]([c]) = f(c) for all cycles c ∈ Cn and all cocycles f ∈ Hom(Cn; G).
Remark. The exact sequence is natural but the splitting not. In this case the naturality
means that for every chain homomorphism f : C → D we have commutative
diagram
0 // Ext(Hn−1(C), G) //
Ext(Hn−1f,idG)

Hn
(C; G)
hC
//
Hnf

Hom(Hn(C), G) //
Hom(Hnf,idG)

0
0 // Ext(Hn−1(D), G) // Hn
(D; G)
hD
// Hom(Hn(D), G) // 0
Proof. The free chain complex (Cn, ∂) determines two other chain complexes, the chain
complex of cycles (Zn, 0) and the chain complex of boundaries (Bn, 0). We have the
short exact sequence of these chain complexes
0 → Zn
i
−→ Cn
∂
−→ Bn−1 → 0.
Since Bn−1 is a subgroup of the free Abelian group Cn−1, it is also free and the exact
sequence splits. Since the functor Hom(−, G) is additive, it converts this sequence into
the short exact sequence of cochain complexes
0 → Hom(Bn−1, G)
δ
−→ Hom(Cn, G)
i∗
−→ Hom(Zn, G) → 0.
As in 5.6 we obtain the long exact sequence of cohomology groups of the given cochain
complexes
→ Hom(Zn−1, G) → Hom(Bn−1, G) → Hn
(C; G)
i∗
−→ Hom(Zn, G)
δ∗
−→ Hom(Bn, G) →
Next, one has to realize how the connecting homomorphism δ∗
in this exact sequence
looks like using its definition and the special form of the short exact sequence. The
conclusion is that δ∗
= j∗
where j : Bn → Zn is an iclusion. Now we can reduce the
long exact sequence to the short one
0 →
Hom(Bn−1, G)
Im j∗
−→ Hn
(C; G)
i∗
−→ Ker j∗
→ 0.
We determine Ker j∗
and Hom(Bn−1, G)/ Im j∗
. Consider the short exact sequence
0 → Bn
j
−→ Zn → Hn(C) → 0.
It is a free resolution of Hn(C). Applying the cofunctor Hom(−, G) we get the cochain
complex
0 → Hom(Zn, G)
j∗
−→ Hom(Bn, G) → 0 → 0 → . . .
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from which we can easily compute that
Hom(Hn(C), G) = Ker j∗
, Ext(Hn(C), G) =
Hom(Bn−1, G)
Im j∗
.
This completes the proof of exactness.
We will find a splitting r : Hom(Hn(C); G) → Hn
(C∗
; G). Let g ∈ Hom(Hn(C), G).
We can define f ∈ Hom(Cn, G) such that on cycles c ∈ Zn
f(c) = g([c])
where [c] ∈ Hn(C). f is a cocycle, hence [f] ∈ Hn
(C; G) and h[f]([c]) = f(c) =
g([c]).
In a very similar way one can compute homology groups with coefficients in G using
the tensor product H∗(C) ⊗ G and Tor(H∗(C), G).
Theorem B. If a free chain complex C has homology groups Hn(C), then the homology
groups Hn(C∗; G) of the chain complex Cn ⊗ G are determined by the split short exact
sequence
0 → Hn(C) ⊗ G
l
−→ Hn(C; G) → Tor(Hn−1(C), G) → 0
where l([c] ⊗ g) = [c ⊗ g] for c ∈ Zn(C), g ∈ G.
6.12. Exercise. Compute cohomology of real projective spaces with Z and Z2 coefficients
using the universal coefficient theorem for cohomology.
Exercise. Using again the universal coefficient theorem for cohomology and Theorem
4.4 prove that that for a given CW-complex X the cohomology of the cochain complex
(Hn
(Xn
, Xn−1
; G), dn
)
where dn
is the composition
Hn
(Xn
, Xn−1
; G)
j∗
n
−→ Hn
(Xn
; G)
δ∗
−→ Hn+1
(Xn+1
, Xn
; G)
is isomorphic to H∗
(X; G). See also Theorem 5.14.
CZ.1.07/2.2.00/28.0041
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