A sequence of objects and morphisms ...A → B → C... in an additive or abelian category (or any category with kernels and images) such that at each object the image of the incoming map equals the kernel of the outgoing map; exactness encodes precise algebraic relationships.

Principle

Exactness enforces local conservation of algebraic information: nothing is lost or unaccounted for at each node because incoming images precisely generate the elements annihilated by the next map.

Demonstration

A short exact sequence 0 → A → B → C → 0 means the first map is injective, the second surjective, and A identifies with ker(B→C); it models extension problems and yields long exact sequences in homology after applying derived functors.

Misapplication

Assuming every short exact sequence splits (so B ≅ A ⊕ C) without verifying a splitting map; this mistake ignores nontrivial extension classes that classify inequivalent middle terms.

Consequence

Exact sequences give rise to diagram lemmas (snake lemma, five lemma) and allow the passage to long exact sequences under homological functors, systematically relating invariants of the objects involved.

Reversal

A nonexact sequence where images are merely contained in kernels but not equal lacks the precise balancing property; such inexactness signals hidden torsion or extension phenomena not captured by naive maps.

Boundary

The notion presupposes the existence of kernels and images; in non-additive categories exactness must be interpreted via monomorphisms/epimorphisms or other categorical substitutes, and behaviour can differ.

Semantic Tension

Tension arises between exactness as a pointwise property of sequences and notions like 'exact functor' that preserve exact sequences; a functor can be left- or right-exact without preserving full exactness.

Synthesis

An exact sequence is a chain of morphisms in which each stage transmits exactly the previous image into the next kernel; this local equality organizes extension, decomposition and homological computations across many algebraic contexts.