= Elliptic complex
An <elliptic complex> on a <smooth manifold> is a finite <cochain complex> of <differential operators> between sections of <vector bundles>,
$$
0\longrightarrow\Gamma(E^0)\xrightarrow{D_0}\Gamma(E^1)\xrightarrow{D_1}\cdots\longrightarrow\Gamma(E^m)\longrightarrow0,
$$
with $D_{i+1}D_i=0$, whose <principal symbol> sequence is exact at every nonzero real cotangent vector. Thus for $\xi\in T_x^*M\setminus\{0\}$, the kernel of each symbol map equals the image of the preceding one, including injectivity and surjectivity at the ends.<A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications by M. F. Atiyah and R. Bott>[section 4, definition of elliptic complexes and leading symbols, printed page 452]{ref}
The <de Rham complex> is the basic example. Up to an inessential nonzero scalar convention, the symbol of $d$ is $\alpha\mapsto\xi\wedge\alpha$. Choose a tangent vector $v$ with $\xi(v)=1$. The <interior product> identity
$$
\iota_v(\xi\wedge\alpha)+\xi\wedge\iota_v\alpha=\alpha
$$
gives a contracting homotopy for that symbol sequence and proves exactness.
The condition belongs to the whole sequence: the individual maps need not be elliptic operators between equal-rank bundles. On a compact manifold without boundary, ellipticity gives finite-dimensional cohomology. Boundary problems require additional conditions. This finiteness is what makes an alternating cohomological trace meaningful in the <Atiyah-Bott fixed-point theorem>.
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