Elliptic complex

This article or section of article is sourced from:Wikipedia

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 $P_{i} : Γ (E_{i}) \to Γ (E_{i + 1})$ 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:

$\dots π^{*} (E_{i - 1}) \to^{σ (P_{i - 1})} π^{*} (E_{i}) \to^{σ (P_{i})} π^{*} E_{i + 1} \dots$