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.
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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.