Proposition 9

Tag 0DX9, permanent. Cite it as [Bourbaki, Tag 0DX9].

Let $g : N \to N_1$ be a homomorphism of $A$ -modules and

\begin{array}{cccccc} (\mathscr{F}_1) & 0 \to M'_1 \xrightarrow{r_1} M_1 \xrightarrow{s_1} M''_1 \to 0 \\ & f' \downarrow & f \downarrow & f'' \downarrow \\ (\mathscr{F}) & 0 \to M' \xrightarrow{r} M \xrightarrow{s} M'' \to 0 \end{array}

a commutative diagram of $A$ -modules with exact rows. The diagram of $k$ -modules

\begin{array}{ccc} \operatorname{Ext}_A(M', N) & \xrightarrow{\delta(\mathscr{F}, N)} & \operatorname{Ext}_A(M'', N) \\ \operatorname{Ext}_A(f', g) \downarrow & & \operatorname{Ext}_A(f'', g) \downarrow \\ \operatorname{Ext}_A(M'_1, N_1) & \xrightarrow{\delta(\mathscr{F}_1, N_1)} & \operatorname{Ext}_A(M''_1, N_1) \end{array}

is commutative.

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

Cited by

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