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A sequence of module homomorphisms ··· → Mᵢ₋₁ →^{fᵢ₋₁} Mᵢ →^{fᵢ} Mᵢ₊₁ → ··· is exact at Mᵢ if the image of fᵢ₋₁ equals the kernel of fᵢ. A short exact sequence 0 → A → B → C → 0 encodes the idea that B is "built from" A and C, with A embedded as a submodule and C isomorphic to the quotient B/A. Exact sequences are the primary language for expressing structural relationships between modules.
In linear algebra, the rank-nullity theorem says that for a linear map T: V → W, dim(V) = dim(ker T) + dim(im T). This is a statement about the exact sequence 0 → ker(T) → V → im(T) → 0. Exact sequences generalize this framework to modules over arbitrary rings, where dimension is unavailable and the structural relationships between modules are more subtle.
A sequence of R-module homomorphisms ··· → Mᵢ₋₁ →^{fᵢ₋₁} Mᵢ →^{fᵢ} Mᵢ₊₁ → ··· is exact at Mᵢ if im(fᵢ₋₁) = ker(fᵢ): the image of the incoming map is exactly the kernel of the outgoing map. The most important case is the short exact sequence 0 → A →^f B →^g C → 0. Exactness at A says ker(f) = 0, so f is injective. Exactness at C says im(g) = C, so g is surjective. Exactness at B says im(f) = ker(g), so A embeds into B and C ≅ B/f(A). In short: B is an extension of C by A.
A short exact sequence 0 → A → B → C → 0 splits if B ≅ A ⊕ C via maps compatible with f and g. This happens if and only if there exists a section s: C → B with g ∘ s = id_C, or equivalently, a retraction r: B → A with r ∘ f = id_A. Over a field, every short exact sequence splits (because every vector space is free), so the theory of extensions is trivial. Over general rings, non-split extensions are ubiquitous — the sequence 0 → ℤ →^{×2} ℤ → ℤ/2ℤ → 0 does not split — and classifying extensions is a central problem solved by the Ext functor.
Exact sequences are not just notation — they are the primary computational tool of homological algebra. Given a functor F (like localization, tensor product, or Hom), you apply it to an exact sequence and ask whether exactness is preserved. Left exact functors (like Hom(−, N)) preserve exactness on the left; right exact functors (like − ⊗ N) preserve it on the right. The failure of full exactness is measured by derived functors: Ext measures the failure of Hom's exactness, and Tor measures the failure of tensor product's exactness. This machinery is the engine of modern commutative algebra and algebraic geometry.