In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient

Ker d / Im d

where d as usual denotes the differentials, indexes skipped for simplicity.

How can this be generalized to a general Abelian Category? Do I have the notion of a Quotient there?

First an answer your question about quotients: One of the axioms of an abelian category says that every morphism has a cokernel. The quotient $B/A$ of a monomorphism $f:A \to B$ is simply its cokernel.

[Recall that the cokernel $p: B \to B/A$ is defined by the following universal property: Given a morphism $g:B \to X$ whose composition with $A \to B$ is zero there is a unique factorization $h: B/A \to X$ such that $g = hp$.]



A morphism $f: A \to B$ in an abelian category has four associated objects and five associated morphisms:



The kernel $\text{ker}\,(f): \text{Ker}\,(f) \to A$, the cokernel $\text{coker}\,(f): B \to \text{Coker}\,(f)$, the coimage $\text{Coim}\,{(f)} = \text{Coker}\,(\text{ker}\,(f))$ and the image $\text{Im}\,(f) = \text{Ker}\,(\text{coker}\,(f))$. The main axiom of abelian categories states that the canonical morphism $\hat{f}:\text{Coim}\,(f) \to \text{Im}\,(f)$ (uniquely determined by requiring that the diagram be commutative) always is an isomorphism.

Now given two morphisms $f:A' \to A$ and $g:A \to A''$ such that $gf = 0$ there are three ways to define the homology of the "complex" $A' \to A \to A''$:


$\text{Coker}\,(\text{Im}\,(f) \to \text{Ker}\,(g))$,
$\text{Ker}\,(\text{Coker}\,(f) \to \text{Coim}\,(g))$,
$\text{Im}\,(\text{Ker}\,(g)\to \text{Coker}\,(f))$.




The first of these corresponds to the usual $\text{Ker}/\text{Im}$ and it is not very hard to show that all three ways give canonically isomorphic objects in an abelian category. It is essential to require the category to be abelian here, the three possibilities are distinct in a general additive category (with kernels and cokernels).
