# An exact sequence bounded by zeros, the alternating sum of the ranks is 0

Note that for an exact sequence
\begin{equation}
0 \to X \to Y \to Z \to 0
\end{equation}

of finitely generated Abelian groups $X$, $Y$, $Z$, we have the result $\operatorname{rank}(Y) = \operatorname{rank}(X) + \operatorname{rank}(Z)$.

First. Decompose the exact sequence you start with into many short exact sequences, each one corresponding to the kernel and image of the maps in the first one.

For example, if $f: A \to B$ is a map, you can construct short exact sequences
\begin{equation}
0 \to \ker f \to A \to \operatorname{im} f \to 0.
\end{equation}

Later. The point of the hint was for you to use the observation provided by your notes themselves. Suppose that we have an exact complex
\begin{equation}
0 \xrightarrow{f_{-1}} X_0 \xrightarrow{f_0} X^1 \xrightarrow{f_1} X^2 \xrightarrow{f_2} \dots \xrightarrow{f_{n-1}} X_n \xrightarrow{f_n} 0
\end{equation}For each $i \in \{-1, \ldots, n\}$, we have the map $f_i: X_i \to X_{i+1}$ (letting $X_{-1} = X_{n+1} = 0$ for convenience) so we have a short exact sequence
\begin{equation}
0 \to \ker f_i \to X_i \to \operatorname{im} f_i \to 0.
\end{equation}

So, assuming we know that the rank is additive in short exact sequences, we have $\operatorname{rk} X_i = \operatorname{rk} \ker f_i + \operatorname{rk} \operatorname{im} f_i$.

Multiplying this by $(-1)^i$ and summing over $i$ we see then that $\sum_{i=-1}^{n} (-1)^i \operatorname{rk} X_i = \sum_{i=-1}^{n} (-1)^i \operatorname{rk} \ker f_i + \sum_{i=-1}^{n} (-1)^i \operatorname{rk} \operatorname{im} f_i.\quad$ (1)

Now, the original exact sequence being exact, we have $\ker f_{i+1} \cong \operatorname{im} f_i$ for all $i$, so of course $\operatorname{rk} \ker f_{i+1} = \operatorname{rk} \operatorname{im} f_i$ for all $i$. Using this in the right-hand side of (1) we easily see that, in fact,$\sum_{i=-1}^{n} (-1)^i \operatorname{rk} X_i = 0.$

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https://hackmd.io/ir8dZ4wBRyeiJ0rGKGx6uw

Introduction to the concept of a topological space and fundamental notions in space theory.

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