Mapping Cone (homological Algebra) - Definition

Definition

The cone may be defined in the category of chain complexes over any additive category (i.e., a category whose morphisms form abelian groups and in which we may construct a direct sum of any two objects). Let be two complexes, with differentials i.e.,

and likewise for

For a map of complexes we define the cone, often denoted by or to be the following complex:

on terms,

with differential

(acting as though on column vectors).

Here is the complex with and . Note that the differential on is different from the natural differential on, and that some authors use a different sign convention.

Thus, if for example our complexes are of abelian groups, the differential would act as

\begin{array}{ccl} d^n_{C(f)}(a^{n + 1}, b^n) &=& \begin{pmatrix} d^n_{A} & 0 \\ f^n & d^n_B \end{pmatrix} \begin{pmatrix} a^{n + 1} \\ b^n \end{pmatrix} \\ &=& \begin{pmatrix} - d^{n + 1}_A & 0 \\ f^{n + 1} & d^n_B \end{pmatrix} \begin{pmatrix} a^{n + 1} \\ b^n \end{pmatrix} \\ &=& \begin{pmatrix} - d^{n + 1}_A (a^{n + 1}) \\ f^{n + 1}(a^{n + 1}) + d^n_B(b^n) \end{pmatrix}\\ &=& \left(- d^{n + 1}_A (a^{n + 1}), f^{n + 1}(a^{n + 1}) + d^n_B(b^n)\right). \end{array}

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