Corollary 2

Tag 0DZQ, permanent. Cite it as [Bourbaki, Tag 0DZQ].

Algebra, chapter X, § 7. Produit de composition

Let

0 \to N \xrightarrow{f_{n-1}} R_n \xrightarrow{f_n} R_{n-1} \to \cdots \xrightarrow{f_2} R_1 \xrightarrow{f_1} M \to 0

be an exact sequence, $\theta \in \mathrm{Ext}_A^n(M, N)$ its associated class, $a_1, \ldots, a_{n+1}$ invisible elements of $k$ . The associated class of the exact sequence

0 \to N \xrightarrow{a_{n+1} f_{n+1}} R_n \xrightarrow{a_n f_n} R_{n-1} \to \cdots \xrightarrow{a_2 f_2} R_1 \xrightarrow{a_1 f_1} M \to 0

is $(a_1^{-1}\, a_2^{-1}\, \ldots\, a_{n+1}^{-1})\ \theta$ .

Indeed, there is a commutative diagram

\begin{array}{ccccccccccccccc} 0 & \rightarrow & N & \xrightarrow{a_{n+1} f_{n+1}} & R_n & \rightarrow & \cdots & \rightarrow & R_2 & \xrightarrow{a_2 f_2} & R_1 & \xrightarrow{a_1 f_1} & M & \rightarrow & 0 \\ & & \downarrow a_1 \ldots a_{n+1} & & \downarrow a_1 \ldots a_n & & & \downarrow a_1 a_2 & & \downarrow a_1 & & \downarrow 1 \\ 0 & \rightarrow & N & \xrightarrow{f_{n+1}} & R_n & \rightarrow & \cdots & \rightarrow & R_2 & \xrightarrow{f_2} & R_1 & \xrightarrow{f_1} & M & \rightarrow & 0 . \end{array}

and we apply the proposition.

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

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