Corollaire 2

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

Algebra, chapter X, § 7. Produit de composition

Consider two short exact sequences of left A-modules

\begin{align*} 0 &\to M' \to M \to M'' \to 0 \\ 0 &\to N' \to N \to N'' \to 0 . \end{align*}

Then the composite connecting homomorphisms

\operatorname{Ext}_A^n(M',N'') \to \operatorname{Ext}_A^{n+1}(M'',N'') \to \operatorname{Ext}_A^{n+2}(M'',N')

and

\operatorname{Ext}_A^n(M',N'') \to \operatorname{Ext}_A^{n+1}(M',N') \to \operatorname{Ext}_A^{n+2}(M'',N')

are opposite.

Indeed, let $\theta_1$ , $\theta_2$ be the classes associated with the given exact sequences, and let $\alpha \in \operatorname{Ext}_A^n(M', M'')$ . The images of $\alpha$ are respectively

\theta_2 \circ \bigl((-1)^{n+1} \alpha \circ \theta_1\bigr) \quad \text{and} \quad (\theta_2 \circ \alpha) \circ \bigl((-1)^{n+2} \theta_1\bigr).

Consider a short exact sequence of left A-modules

(\mathscr{S}) \qquad 0 \to N \to R_n \xrightarrow{f_n} R_{n-1} \xrightarrow{f_{n-1}} \cdots \to R_1 \xrightarrow{f_1} M \to 0

and set $K_0=M$ , $K_i=\operatorname{Ker}f_i$ , $i=1,\ldots,n-1$ , $K_n=N$ . We therefore have exact sequences

(9)\qquad 0\to K_i\to R_i\to K_{i-1}\to0,\qquad 1\leq i\leq n,

to which are associated for every $A$ left module $P$ , connecting homomorphisms

\operatorname{Ext}_A^m(P,K_{i-1})\to\operatorname{Ext}_A^{m+1}(P,K_i),

\operatorname{Ext}_A^m(K_i,P)\to\operatorname{Ext}_A^{m+1}(K_{i-1},P),

whence by composition of the iterated connecting homomorphisms, associated with $(\mathscr{S})$

\delta^m(P,\mathscr{S}):\operatorname{Ext}_A^m(P,M)\to\operatorname{Ext}_A^{m+n}(P,N)

\delta^m(\mathscr{S},P):\operatorname{Ext}_A^m(N,P)\to\operatorname{Ext}_A^{m+n}(M,P).

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

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