---
tag: M8GMH8GQR6
kind: concept
name: Elliptic complex
msc: [58J10, 58J20]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_complex
same_as:
  wikidata: Q5365791
  wikipedia: Elliptic complex
---

## Summary

An elliptic complex is a sequence of [[differential operators between vector bundles|CGBX1N9V8H]] whose consecutive composites vanish.
Its [[principal symbols|S8GC34D5ZG]] must form an exact sequence at each nonzero covector.
It generalises an [[elliptic operator|MT9P2X85V8]], which is an elliptic complex of length two.
On a compact manifold its cohomology groups are finite-dimensional, and its index is their alternating sum of dimensions.
The [[de Rham complex|GX74C07T0G]] and the Dolbeault complex are the basic examples.

## Explanations

### L1 human

Some important operators come in chains, where applying two consecutive operators always gives zero, as $d \circ d = 0$ for the exterior derivative.
Then one studies the cohomology: what the next operator kills, modulo what the previous one produces.
An elliptic complex is a chain of this kind that is elliptic in the sense that, at top order and in every nonzero frequency direction, the chain is exact.
This makes the cohomology finite-dimensional on a closed manifold.
For the de Rham complex, this cohomology is the de Rham cohomology, and its alternating sum of dimensions is the Euler characteristic.