# Elliptic complex

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Let $M$ be a differential manifold and $E_i$ be smooth vector bundles over $M$. Let $P_i:\Gamma(E_i) \to \Gamma(E_{i+1})$ form a differential complex (viz $P_i \circ P_{i-1} = 0$. Then this differential complex is said to be elliptic if the following sequence of symbols is exact:
$\ldots \pi^*(E_{i-1}) \to^{\sigma(P_{i-1})} \pi^*(E_i) \to^{\sigma(P_i)} \pi^*{E_{i+1}} \ldots$