Corollary 1

Tag 0541, permanent. Cite it as [Bourbaki, Tag 0541].

Commutative Algebra, chapter VII, § 4. Modules over integrally closed Noetherian domains

Let $0 \to M, \xrightarrow{u} M_{n-1} \to \cdots \to M_0 \to 0$ be an exact sequence of finitely generated A-modules. Then

\sum_{i=0}^n (-1)^i c(M_i) = 0.

We argue by induction on $n$ , the case $n = 2$ being Proposition 16 (i). If $M'_{n-1} = \operatorname{Coker}(u)$ , there are the two exact sequences:

\begin{align*} 0 &\to M, \to M_{n-1} \to M'_{n-1} \to 0 \\ 0 &\to M'_{n-1} \to M_{n-2} \to \cdots \to M_0 \to 0. \end{align*}

The first shows that $M'_{n-1}$ is finitely generated and the induction hypothesis gives

(-1)^{n-1} c(M'_{n-1}) + \sum_{t=0}^{n-a} (-1)^t c(M_t) = 0

and

c(M'_{n-1}) = c(M_{n-1}) - c(M_n),

whence the corollary.

An exact sequence

0 \to L_n \to L_{n-1} \to \cdots \to L_0 \to E \to 0

where the $L_i$ ( $0 \leq i \leq n$ ) are finitely generated free A-modules, is called a finite free resolution of the A-module E.

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

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Cites

Proposition 16