Let $C_*$ be a chain complex such that each $C_i$ is a torsion-free, finite-range abelian group with $C_i=0$ for all $i<0$. Suppose that $C_i=0$ for all $i$ is large enough. The Euler characteristic of $C_*$ chain complex is defined as $$\chi(C_*)=\sum_{i\geq 0}(-1)^iRank(C_i)$$ Prove that $$\chi(C_*)=\sum_{i\geq 0}(-1)^iRank(H_i(C_*))$$
I have to prove that $\sum_{i\geq 0}(-1)^iRank(C_i)=\sum_{i\geq 0}(-1)^iRank(H_i(C_*))$, I think that one way to do this is by showing that $Rank(C_i)=Rank(H_i(C_*))$ but I do not know if this is true in general, in a nutshell, I want to show that the cardinality of the base of any $C_i$ is the same as the cardinality of the basis of the corresponding homology, how can I do this? Thank you
Best Answer
$\newcommand{\rank}{\operatorname{rk}}$ Consider the short exact sequences for $i\in \mathbb Z$.
$$0\to Z_i \to C_i \stackrel{d}\to B_{i-1} \to 0$$ $$0\to B_i \to Z_i \to H_i \to 0$$
From them it follows that for each $i\in \mathbb Z$, $$ \rank C_i=\rank Z_i + \rank B_{i-1}$$ $$\rank H_i = \rank Z_i-\rank B_i$$
Then
\begin{align*} \sum_{i\in\mathbb Z}(-1)^i \rank H_i&= \sum_{i\in\mathbb Z}(-1)^i \rank Z_i-\sum_{i\in\mathbb Z}(-1)^i \rank B_i\\ &=\sum_{i\in\mathbb Z}(-1)^i \rank Z_i+\sum_{i\in\mathbb Z}(-1)^{i-1} \rank B_i\\ &=\sum_{i\in\mathbb Z}(-1)^i \rank C_i. \end{align*}
This proof works in general for any map $\chi$ from some class of modules to integers which is additive on short exact sequences, meaning that whenever $$0\to A\to B\to C\to 0$$ is exact, $\chi(A) + \chi(C) = \chi(B)$.