Corollary 2

Tag 05RZ, permanent. Cite it as [Bourbaki, Tag 05RZ].

Let $(E_i)_{0 \leq i \leq n}$ be a finite family of vector spaces over a field $K$ . If there exists an exact sequence of linear mappings

0 \longrightarrow E_0 \xrightarrow{u_0} E_1 \xrightarrow{u_1} E_2 \longrightarrow \cdots \longrightarrow E_{n-1} \xrightarrow{u_{n-1}} E_n \xrightarrow{u_n} 0

the relation

\sum_{2k+1 \leq n} \dim E_{2k+1} = \sum_{2k \leq n} \dim E_{2k}

holds, or, if all the spaces are finite dimensional,

\sum_{i=1}^n (-i)^i \dim E_i = 0.

Let $I_k = \operatorname{Im}\ u_k = \operatorname{Ker}\ u_{k+1}$ for $0 \leq k \leq n-1$ ; $I_{k+1}$ is therefore isomorphic to $E_{k+1}/I_k$ , hence (formula (3)) $\dim I_k + \dim I_{k+1} = \dim E_{k+1}$ for $0 \leq k \leq n-2$ and moreover $\dim I_0 = \dim E_0$ and $I_{n-1} = E_n$ , hence $\dim I_{n-1} = \dim E_n$ . Replacing $\dim E_i$ by its expression as a function of the $\dim I_k$ in the two sides of (5), we obtain on each side $\sum_{k=0}^{n-1} \dim I_k$ , whence the corollary.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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