Remark

Tag 0E0C, permanent. Cite it as [Bourbaki, Tag 0E0C].

Algebra, chapter X, § 7. Produit de composition

Via the commutation isomorphisms, we deduce from a) an analogous statement in the case of an exact sequence (16) of right A-modules.

Let $b : M' \to E'$ be a right resolution of $M'$ ; we thus have an exact sequence

0 \to M' \xrightarrow{b^0} {E'}^0 \xrightarrow{\delta^0} {E'}^1 \to \ldots \to {E'}^k \xrightarrow{\delta^k} {E'}^{k+1}

corresponding to a right resolution $E$ of $M$ ; let $\sigma : E'(n) \to E$ be the morphism such that $\sigma^k = 1_{{E'}^{k-n}}$ for $k \geq n$ . We thus have homomorphisms

H(\mathrm{Homgr}_A(1_N, \sigma)) : H(\mathrm{Homgr}_A(N, E'))(n) \to H(\mathrm{Homgr}_A(N, E)) .

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

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