Proposition 4

Tag 0DY6, permanent. Cite it as [Bourbaki, Tag 0DY6].

Let M be a left A-module, $a : R \to M$ a left resolution of M such that the sequence

(7)

0 \rightarrow \mathrm{Homgr}_A(R, E') \xrightarrow{\mathrm{Homgr}(\tilde{s}, 1)} \mathrm{Homgr}_A(R, E) \xrightarrow{\mathrm{Homgr}(\tilde{r}, 1)} \mathrm{Homgr}_A(R, E'') \rightarrow 0

is exact. Then the following diagram is commutative.

\begin{array}{ccc} \mathrm{H}(\mathrm{Homgr}_A(R, E'')) & \xrightarrow{\partial((7))} & \mathrm{H}(\mathrm{Homgr}_A(R, E')) \\ \varphi(R, E'') \downarrow & & \downarrow \varphi(R, E') \\ \mathrm{Ext}_A(M, N'') & \xrightarrow{\delta(M, (5))} & \mathrm{Ext}_A(M, N') \end{array}

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

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