What does $d_{n+1}\circ{d_{n}}=0$ mean in the definition of a chain complex?

Question

According to the Wiki article on chain complexes, a chain complex $(A_{\bullet},d_{\bullet})$ is a sequence of abelian groups or modules connected by homomorphisms such that $d_{n+1}\circ{d_{n}}=0$. Does this mean that given an element $a\in{A_{n+1}}$ that $(d_{n+1}\circ{d_{n}})(a)=d_{n+1}(d_{n}(a))=0$ or am I mistaken? I realize that the question is poorly asked but any elucidation on the meaning of the composition of the homomorphisms being equal to zero would be much appreciated.

Answer 1

You are correct.

We have $d_{n+1} : A_{n+1} \to A_n$ and $d_n : A_n \to A_{n-1}$, so we can compose them to obtain a map $d_{n+1}\circ d_n : A_{n-1} \to A_{n-1}$. The condition states that the map $d_{n+1}\circ d_n$ is the zero map (zero is used to denote the identity of $A_{n-1}$). In particular, for every $a \in A_{n+1}$, $(d_{n+1}\circ d_n)(a) = 0$.