NOTES ON BASIC HOMOLOGICAL ALGEBRA

NOTES ON BASIC HOMOLOGICAL ALGEBRA




1. Chain complexes and their homology
Let R be a ring and ModR the category of right R-modules; a very similar discussion can be
had for the category of left R-modules R Mod also makes sense and is left to the reader. Then
a sequence of R-module homomorphisms
f g
L−
→M −
→N
is said to be exact if Ker g = Im f . Of course this implies that gf = 0. More generally, a
sequence of homomorphisms
fn+1 fn fn−1
· · · −−−→ Mn −→ Mn−1 −−−→ · · ·
is exact if for each n, the sequence
fn+1 fn
Mn+1 −−−→ Mn −→ Mn−1
is exact. An exact sequence of the form
f g
0→L−
→M −
→N →0
is called short exact. Such a sequence is split exact if there is a homomorphism r : M −→ L (or
equivalently j : N −→ M ) so that rf = idL (respectively gj = idN ).

w; LO
id wwww
ww r
ww g
0→L /M /N →0
O ;
f vv
j vvv
vv
vv id
N
These equivalent conditions imply that M ∼ = L ⊕ N . Such homomorphisms r and g are said to
be retractions, while L and N are said to be retracts of M .
A sequence of homomorphisms
dn+1 n d dn−1
· · · −−−→ Cn −→ Cn−1 −−−→ · · ·
is called a chain complex if for each n,
(1.1) dn dn+1 = 0,

or equivalently,

Im dn+1 ⊆ Ker dn .
An exact or acyclic chain complex is one which each segment
dn+1 n d
Cn+1 −−−→ Cn −→ Cn−1

Date: [28/02/2009].
1

, 2 ANDREW BAKER

is exact. We write (C∗ , d) for such a chain complex and refer to the dn and boundary homo-
morphisms. We symbolically write d2 = 0 to indicate that (1.1) holds for all n. For clarity we
sometimes (C∗ , dC ) to indicate which boundary is being used.
If a chain complex is of finite length finite we often pad it out to a doubly infinite complex
by adding in trivial modules and homomorphisms. In particular, if M is a R-module we can
view it as the chain complex with M0 = M and Mn = 0 whenever n 6= 0. It is often useful to
consider the null complex 0 = ({0}, 0).
(C∗ , d) is called bounded below if there is an n1 such that Cn = 0 whenever n < n1 . Similarly,
(C∗ , d) is called bounded above if there is an n2 such that Cn = 0 whenever n > n2 . (C∗ , d) is
called bounded if it is bounded both below and above.
Given a complex (C∗ , d), its homology is defined to be the complex (H∗ (C∗ , d), 0) where
Hn (C∗ , d) = Ker dn / Im dn+1 .
The homology of a complex measures its deviation from exactness; in particular, (C∗ , d) is exact
if and only if H∗ (C∗ , d) = 0. Notice that there are exact sequences
0 → Im dn+1 −−−−→ Ker dn −−−−→ Hn (C∗ , d) → 0,
(1.2)
0 → Ker dn −−−−→ Cn −−−−→ Im dn → 0.
Example 1.1. Consider the complex of Z-modules where
Cn = Z/4, d : Z/4 −→ Z/4; d(t) = 2t.
Then Ker dn = 2 Z/4 = Im dn and Hn (C∗ , d) = 0, hence (C∗ , d) is acyclic.
A homomorphism of chain complexes or chain homomorphism h : (C∗ , dC ) −→ (D∗ , dD ) is a
sequence of homomorphisms hn : Cn −→ Dn for which the following diagram commutes.
dC
Cn −−−n−→ Cn−1
 
 h
hn y y n−1
dD
Dn −−−n−→ Dn−1
We often write h : C∗ −→ D∗ when the boundary homomorphisms are clear from the context.
A chain homomorphism for which each hn : Cn −→ Dn is an isomorphism is called a chain
isomorphism and admits an inverse chain homomorphism D∗ −→ C∗ consisting of the inverse
homomorphisms h−1 n : D∗ −→ C∗ .
The category of chain complexes in ModR , ChZ (ModR ), has chain complexes as its objects
and chain homomorphisms as its morphisms. Like ModR , it is an abelian category.
Let h : (C∗ , dC ) −→ (D∗ , dD ) be a chain homomorphism. If u ∈ Ker dC
n we have

dD C
n (hn (u)) = hn (dn (u)) = 0,

and if v ∈ Cn+1 ,
hn (dC D
n+1 (v)) = dn+1 (hn+1 (v)).
Together these allow us to define for each n a well-defined homomorphism
h∗ = Hn (h) : Hn (C∗ , dC ) −→ Hn (D∗ , dD ); h∗ (u + Im dC D
n+1 ) = hn (u) + Im dn+1 .

If g : (B∗ , dB ) −→ (C∗ , dC ) is another morphism of chain complexes, it is easy to check that
(hg)∗ = h∗ g∗ ,

while for the identity morphism id : (C∗ , dC ) −→ (C∗ , dC ) we have

id∗ = id .
Thus each Hn is a covariant functor from chain complexes to R-modules.