Differential complex

Definition

Given data

Let $M$ be a differential manifold and $E_i$s be smooth vector bundles over $M$. For a vector bundle $E$, let $\mathrm{\Gamma }(E)$ dneote the shead of smooth sections of $E$.

Definition part

A differential complex is a sequence of maps $P_i:\mathrm{\Gamma }(E_i)\to \mathrm{\Gamma }(E_{i+1})$ such that $P_i\circ P_{i-1}=0$.