Definition A chain complex C_∗ is a (doubly infinite) sequence (C_i: i ∈ Z) of modules over some unital ring, together with homomorphisms ∂_I : C_i → C_i−1 for each i ∈ Z, such that ∂_i ◦ ∂_i+1 = 0 for all integers i.

The ith homology group H_i(C_∗) of the complex C_∗ is defined to be the quotient group Z_i(C_∗)/B_i(C_∗), where Z_i(C_∗) is the kernel of ∂_i: C_i → C_i−1 and B_i(C_∗) is the image of ∂_i+1: C_i+1 → C_i .

Note that if the modules C_∗ occuring in a chain complex C_∗ are modules over some unital ring R then the homology groups of the complex are also modules over this ring R.

Definition Let C_∗ and D_∗ be chain complexes. A chain map f: C_∗ → D_∗ isa sequence f_i: C_i → D_i of homomorphisms which satisfy the commutativity condition ∂_i ◦ f_i = f_i−1 ◦ ∂_ifor all i ∈ Z.

Note that a collection of homomorphisms f_i : C_i → D_i defines a chain map f_∗: C_∗ → D_∗ if and only if the diagram

is commutative.

Let C_∗ and D_∗ be chain complexes, and let f_∗: C_∗ → D_∗ be a chain map.

Then f_i(Z_i(C_∗)) ⊂ Z_i(D_∗) and f_i(B_i(C_∗)) ⊂ B_i(D_∗) for all i. It follows from this that f_i : C_i → D_i induces a homomorphism f_∗: H_i(C_∗) → H_i(D_∗) of homology groups sending [z] to [f_i(z)] for all z ∈ Z_i(C_∗), where [z] = z + B_i(C_∗), and [f_i(z)] = f_i(z) + B_i(D_∗).

Definition A short exact sequence 0−→A_∗  ^p∗→B_∗ ^q∗→C_∗→0 of chain complexes consists of chain complexes A_∗, B_∗ and C_∗ and chain maps p_∗: A_∗ → B_∗ and q_∗: B_∗ → C_∗ such that the sequence

0→A_i    ^pi →B_i  ^qi→C_i→0

is exact for each integer i.

We see that 0→A_∗^p∗ −→B_∗^q∗→C_∗→0 is a short exact sequence of chain complexes if and only if the diagram

Lemma 1.1 Given any short exact sequence 0→A_∗^p∗→B_∗^q∗→C_∗→0 of chain complexes, there is a well-defined homomorphism

                               α_i: H_i(C_∗) → H_i−1(A_∗)

which sends the homology class [z] of z ∈ Z_i(C_∗) to the homology class [w] of any element w of Z_i−1(A_∗) with the property that p_i−1(w) = ∂_i(b) for some b ∈ B_i satisfying q_i(b) = z.

Proof Let z ∈ Z_i(C_∗). Then there exists b ∈ Bi satisfying q_i(b) = z, since

q_i: B_i → C_i   is surjective. Moreover

                         q_i−1(∂_i(b)) = ∂_i(q_i(b)) = ∂_i(z) = 0.

But p_i−1: A_i−1 → B_i−1 is injective and p_i−1(A_i−1) = ker q_i−1, since the sequence

                                 0−→A_i−1   ^pi−1−→B_i−1 ^qi−1→C_i−1

is exact. Therefore there exists a unique element w of A_i−1 such that ∂_i(b) =p_i−1(w). Moreover

                     p_i−2(∂_i−1(w)) = ∂_i−1(p_i−1(w)) = ∂_i−1(∂_i(b)) = 0

(since ∂_i−1 ◦ ∂_i = 0), and therefore ∂_i−1(w) = 0 (since p_i−2: A_i−2 → B_i−2 is injective). Thus w ∈ Z_i−1(A_∗).