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A chain complex in an abelian category A is a sequence of objects and morphisms ··· → C_{n+1} → C_n → C_{n-1} → ··· where the composition of any two consecutive morphisms (called boundary or differential maps, d_{n} ∘ d_{n+1} = 0) is zero. A sequence is exact at C_n if the image of d_{n+1} equals the kernel of d_n, meaning "what goes in as boundaries is exactly what would be killed." A short exact sequence 0 → A → B → C → 0 captures the idea that A embeds in B and C is the quotient B/A. Chain complexes and their morphisms form an abelian category Ch(A), enabling the systematic study of homological invariants.
Work with chain complexes of abelian groups. Construct the short exact sequence 0 → Z →(×2) Z → Z/2Z → 0 and verify exactness at each position. Then build a longer chain complex, compute where it fails to be exact, and observe that the failure is measured by homology groups. Understand the chain map between two complexes and verify it preserves the differential.
A chain complex is a sequence of objects linked by maps — called boundary or differential maps — with one crucial constraint: the composition of any two consecutive maps is zero. Written as d ∘ d = 0, this says that whatever the first map sends into the middle object lies entirely in the kernel of the next map. If you think of the differentials as "boundary operators" in a geometric context, d ∘ d = 0 encodes the intuition that the boundary of a boundary is always empty. In the algebraic setting it simply means im(d_{n+1}) ⊆ ker(d_n) at every position.
Exactness is a stronger condition than d ∘ d = 0. A complex is exact at object C_n if im(d_{n+1}) = ker(d_n) — not just a subset, but equality. An exact sequence has no "holes": every element killed by the outgoing map was the image of something from the incoming map. Short exact sequences 0 → A → B → C → 0 are the building blocks of this theory. Exactness at A means the first map is injective (A embeds into B); exactness at C means the second map is surjective; exactness at B means the image of A in B is precisely the kernel of the map to C, so C ≅ B/A. These sequences encode extension problems: given A and C, how many non-isomorphic ways can B be built?
Homology is the tool for measuring how far a complex deviates from exactness. The n-th homology group is defined as H_n = ker(d_n)/im(d_{n+1}). Elements of ker(d_n) are called cycles; elements of im(d_{n+1}) are called boundaries. Since d ∘ d = 0 guarantees every boundary is a cycle, this quotient is well-defined. When H_n = 0 at every position, the complex is exact everywhere. When H_n ≠ 0, the non-trivial elements are cycles that are not boundaries — they detect "holes" that the differential cannot reach.
A subtle but important point is that short exact sequences do not automatically split. Splitting would mean B ≅ A ⊕ C, with the sequence decomposing into a direct sum. But the sequence 0 → Z →(×2) Z → Z/2Z → 0 is exact, and yet Z cannot be written as Z ⊕ Z/2Z — an infinite cyclic group cannot contain a nontrivial finite cyclic group as a direct summand. The splitting lemma characterizes exactly when splitting occurs (when a right-inverse or left-inverse exists), and failure to split is precisely what makes extension theory rich. Chain complexes and exact sequences are the language in which homological algebra, algebraic topology, and derived functors are all formulated, so mastering these definitions is the gateway to understanding how algebraic invariants detect geometric structure.