Remark 1

Tag 0E03, permanent. Cite it as [Bourbaki, Tag 0E03].

Algebra, chapter X, § 7. Produit de composition

If one uses the isomorphisms $\psi$ of X, p. 69, one can also define the composition product by the commutative diagram

\begin{array}{ccc} \mathrm{Ext}_A^n(P, Q) \times \mathrm{Tor}_m^A(P, M) & \xrightarrow{c_{P,Q;M}} & \mathrm{Tor}_{m-n}^A(Q, M) \\ \bar{a}_{P,Q} \times \psi_{P(M)} \downarrow & & \downarrow \psi_{Q(M)} \\ H^n(\mathrm{Homgr}_A(L(P), L(Q))) \times H_m(L(P) \otimes_A M) & \longrightarrow & H_{m-n}(L(Q) \otimes_A M); \end{array}

in other words, one represents $\alpha$ by a morphism $f$ from $L(P)$ to $L(Q) (-n)$ , $\gamma$ by a cycle $x \in L_m(P) \otimes_A M$ , and $\alpha \circ \gamma$ is the class of the cycle

(f_m \otimes 1_M)(x) \in L_{m-n}(Q) \otimes_A M.

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

Cited by

Nothing.

Cites

Nothing.