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Elliptic complex

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Definition

Symbol-free definition

A differential complex is said to be elliptic if its sequence of symbols is exact.

Definition with symbols

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