Cone of a complex

Assume we are talking about complexes of objects in an additive category $\mathcal{C}$ .

DEFINITION 11.1   [1, 4.1] For any complex $X^\bullet$ , we define $TX^\bullet$ to be a complex defined by

$$(TX)^i= X^{i+1} , \quad d_{T X} = -d_X.$$

DEFINITION 11.2   [1, 4.3] Let $u:X^{\bullet} \to Y^{\bullet}$ be a morphism of complexes. The cone $C_u^\bullet$ of $u$ is defined to be a graded object

$$Y^{\bullet} \oplus T X^{\bullet}$$

equipped with the following differential:

$$d \begin{pmatrix} y \\ x \end{pmatrix}= \begin{pmatrix} d_Y & u \\ 0 & -d_X \end{pmatrix}\begin{pmatrix} y \\ x \end{pmatrix}$$

Idea 1: Instead of considering kernel and cokernel of a morphism $u$ , we consider its cone $C_u$ .

For any $u$ , we have morphisms (triangle):

$$X^\bullet \overset{u}{\to} Y^\bullet \overset {\iota_Y}{\to} C_u^\bullet \overset {p_{T X}}{\to} T X^\bullet.$$

Let us call such a triangle standard. Now if $\mathcal{C}$ is abelian, then for each standard triangle as above we have the following long exact sequence:

$$\dots \to H^k (X^\bullet)\to H^k (Y^\bullet) \to H^k(C_u^\bullet) \to H^{k+1}(X^\bullet)\to \dots$$