Is there a terminological difference between "sequence" and "complex" in homology theory Suppose you are given something like this:
$\dots \longrightarrow A^n \longrightarrow A^{n+1} \longrightarrow \dots$
People tend to talk about "chain complexes" but about "short exact sequences". Is there any terminological difference or any convention with regards to using these words (EDIT: I mean "complex" and "sequence" in a homological context) that a mathematical writer should comply to?

Let $R$ be a ring. A chain complex is a sequence (of left $R$-modules)
$A = \cdots\to A_{n}\xrightarrow{d} A_{n-1}\xrightarrow{d}\cdots$
Such that $d^2 = 0$. This allows you to define $H_n(A) = \ker{d_n}/\mathrm{im}~d_{n+1}$.
An exact complex is a chain complex such that $H_n(A) = 0$ for all $n$. Exact complexes are usually called acyclic complexes.
A short exact sequence is an exact chain complex of the form $0\to A\to B\to C\to 0$. This is equivalent to saying $A\to B$ is an injection, and $B\to C$ is a surjection whose kernel is the image of $A$ under $A\to B$.
