Exact sequence

Summary

An exact sequence is a chain of modules and linear maps in which the image of each map equals the kernel of the next. Short exact sequences $0 \to A \to B \to C \to 0$ describe $B$ as built from a submodule $A$ and a quotient $C$. Exact sequences are the language of homological algebra: one asks which functors, such as $\Hom$ and the tensor product, preserve them. The same notion makes sense for groups and in any abelian category.