# Homotopy category of chain complexes and a example of a null complex

Let $\mathcal{A}=A\text{-mod}$ be an abelian category, where $A$ be a finite-dimensional algebra over a algebraically closed field. Let $K(\mathcal{A})$ be a category whose objects are complexes in $\mathcal{A}$ and whose morphisms are homotopy equivalence classes of morphisms of complexes.

Let $P^\bullet \in K^- (A\text{-proj})$ with $$H^j(P^\bullet)=\frac{\ker \; d_{P^\bullet}^j}{\operatorname{im} d_{P^\bullet}^{j-1}} = 0 \text{ for } j\leq-(n+1).$$ We have an exact sequence of complexes:

Suppose $\operatorname{coker} d^{-(n+1)}$ projective, we have $\operatorname{im} d^{-(n+1)}$ projective by exact sequence complexes above.

Why is the complex $\widetilde{P^\bullet}$ zero in $K^- (A\text{-proj})$?

there is a proposition： If $A$ is an Abelian category with enough projective,then $D^-(A)\cong K^-(A-proj)$.