---
tag: 8722T9KCP5
kind: definition
name: Elliptic complex
msc: [58J10, 58J20]
defines: M8GMH8GQR6
uses: [CGBX1N9V8H, S8GC34D5ZG, MT9P2X85V8, P4DZM975BV, F45VQ47GPV, 0ZHSTAG8ZY, GX74C07T0G, RNDDWDAFJ5]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_complex
---

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]], and let $E_0, E_1, \ldots, E_N$ be smooth [[vector bundles|P4DZM975BV]] over $M$.
For $0 \le k < N$, let $P_k \colon \Gamma(E_k) \to \Gamma(E_{k+1})$ be a [[differential operator|CGBX1N9V8H]] of order $m_k$, with [[principal symbol|S8GC34D5ZG]] $\sigma(P_k) = \sigma_{m_k}(P_k)$.
The sequence
$$0 \to \Gamma(E_0) \xrightarrow{P_0} \Gamma(E_1) \xrightarrow{P_1} \cdots \xrightarrow{P_{N-1}} \Gamma(E_N) \to 0$$
is an *elliptic complex* if both of the following hold.

1. $P_{k+1} \circ P_k = 0$ for $0 \le k < N - 1$.
2. For every $x \in M$ and every nonzero $\xi \in T^*_x M$, the sequence of linear maps
$$0 \to E_{0,x} \xrightarrow{\sigma(P_0)(x, \xi)} E_{1,x} \to \cdots \xrightarrow{\sigma(P_{N-1})(x, \xi)} E_{N,x} \to 0$$
is exact.

## Remarks

For $N = 1$ the complex is a single operator $P_0$, and it is elliptic exactly when $P_0$ is an [[elliptic operator|MT9P2X85V8]].
When $M$ is compact without boundary, the cohomology groups $\ker P_k / \operatorname{im} P_{k-1}$ are finite-dimensional, and the index of the complex is $\sum_k (-1)^k \dim(\ker P_k / \operatorname{im} P_{k-1})$.
Suppose that all $P_k$ have the same order and that $M$ and the bundles carry metrics, and write $P = \sum_k P_k$ and $P^* = \sum_k P_k^*$ for the formal adjoint.
Then $P + P^* \colon \Gamma(\bigoplus_k E_{2k}) \to \Gamma(\bigoplus_k E_{2k+1})$ is elliptic and has the same index as the complex.

Example: the de Rham complex $0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \cdots \xrightarrow{d} \Omega^n(M) \to 0$ of a manifold of dimension $n$, with the [[exterior derivative|0ZHSTAG8ZY]].
It satisfies $d \circ d = 0$, and the symbol of $d$ at $\xi$ is $\omega \mapsto \xi \wedge \omega$.
For $\xi \ne 0$, complete $\xi$ to a basis $\xi = e_1, e_2, \ldots, e_n$ of $T^*_x M$.
A form $\omega$ with $\xi \wedge \omega = 0$ is then a sum of terms containing $e_1$.
So it equals $\xi \wedge \eta$ for some $\eta$, which proves exactness.
Its cohomology is the [[de Rham cohomology|GX74C07T0G]], and its index is the [[Euler characteristic|RNDDWDAFJ5]] of $M$.

Non-example: on a manifold of dimension $n \ge 2$, the two-term complex $0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \to 0$ satisfies clause 1 but not clause 2.
The symbol sequence $0 \to \R \to T^*_x M \to 0$ is not exact at $T^*_x M$, because the image of $c \mapsto c \, \xi$ is a line in a space of dimension $n$.

## Explanations

### L1 human

An elliptic complex is a chain of differential operators that compose to zero and that, at top order in each nonzero direction, fit together exactly.