For a chain map , one mapping-cone convention takes with a differential combining and . Its short exact sequence with and a shift of produces a long exact sequence in homology.
For a chain map , use and . The chain map identity makes . The short exact sequence of chain complexes has connecting homomorphism in degree . Its long exact sequence in homology shows that the homology of vanishes precisely when every is an isomorphism.
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In homological algebra, a mapping cone is a construction that allows us to define a new complex from a given morphism of chain complexes. It plays a significant role in various contexts, such as in the study of derived categories and in the formulation of the Long Exact Sequence in homology.