In Hatcher's Algebriac Topology, on page 190, he mentions that a chain complex of finitely generated abelian groups
always splits as the direct sum of elementary complexes of the forms $0\rightarrow\mathbb{Z}\rightarrow 0$ and $0\rightarrow\mathbb{Z}\xrightarrow{m}\mathbb{Z}\rightarrow 0$.
I understand that this seems to be related to the Smith normal form. But what precisely does it mean that a chain complex is a "direct sum of elementary complexes"? There isn't anything said about the algebra of chain complexes. What am I taking the direct sum of, and how am I obtaining the final boundary map?
Best Answer
If $C_\bullet$ and $D_\bullet$ are chain complexes with boundary maps $\partial_C$ and $\partial_D$ respectively, then their direct sum is defined to be the chain complex with $n$th term $C_n \oplus D_n$, and with boundary map $$ C_n \oplus D_n \xrightarrow{\partial_C \oplus \partial_D} C_{n-1} \oplus D_{n-1}. $$