Remark 1

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

Algebra, chapter X, § 7. Produit de composition

Let (P, p) be a projective resolution of M. By X, p. 49. prop. 3, there exists a commutative diagram

\begin{array}{ccccccccccccc} 0 & \longrightarrow & N & \longrightarrow & R_n & \longrightarrow & \ldots & \longrightarrow & R_1 & \longrightarrow & M & \longrightarrow & 0 \\ & & \uparrow & & \uparrow & & & & \uparrow & & \uparrow & & \uparrow \\ P_n & \longrightarrow & P_{n-1} & \longrightarrow & \ldots & \longrightarrow & P_0 & \xrightarrow{p} & M & \longrightarrow & 0. \end{array}

With the notations of § 6, θ is the image under φ(P, N) of the homotopy class of the morphism P → N defined by (−1)^{n(n+1)/2} \tilde{u}_n. Analogously, if (e, E) is an injective resolution of N, there exists a commutative diagram

\begin{array}{cccccccccccc} 0 & \longrightarrow & N & \longrightarrow & E^0 & \longrightarrow & \ldots \longrightarrow & E^{n-1} & \longrightarrow & E^n \\ & & ^{1_N}\uparrow & & ^{\tilde{v}^0}\uparrow & & & ^{\tilde{v}^{n-1}}\uparrow & & ^{\tilde{v}^n}\uparrow \\ 0 & \longrightarrow & N & \longrightarrow & R_n & \longrightarrow & \ldots \longrightarrow & R_1 & \longrightarrow & M & \longrightarrow & 0 . \end{array}

and θ is the image under φ(M, E) of the homotopy class of the morphism M → E defined by (−1)^{n(n+1)/2} v^n. This follows from the construction of θ and the definitions of φ(P, N) and φ(M, E).

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

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