A long exact sequence is a sequence of abelian groups or modules with arbitrarily many consecutive maps, for which the image of every map equals the kernel of the next. A short exact sequence of cochain complexes yields a long exact sequence of their cohomology, with connecting homomorphisms increasing degree by one. The exact sequences of relative cohomology and the Bockstein homomorphism are examples.
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In mathematics, particularly in the field of algebraic topology and homological algebra, an **exact sequence** is a sequence of algebraic objects (like groups, modules, or vector spaces) connected by morphisms (like group homomorphisms or module homomorphisms) such that the image of one morphism is equal to the kernel of the next. This concept is crucial because it encapsulates the idea of relationships between structures and helps in understanding their properties.