Definition 2

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

Algebra, chapter X, § 4. Produits de torsion

The linking homomorphism of torsion products, relative to the module $M$ and the exact sequence $\mathscr{E}$ , is the composite homomorphism

\partial(M, \mathscr{E}) = \psi_M(N')^{-1} \circ \partial^{(M\mathscr{E})} \circ \psi_M(N'') : \mathrm{Tor}^A(M, N'') \to \mathrm{Tor}^A(M, N')

This is a graded $k$ -homomorphism of degree $-1$ , whose homogeneous components are denoted $\partial_n(M, \mathscr{E}) : \mathrm{Tor}_n^A(M, N'') \to \mathrm{Tor}_{n-1}^A(M, N')$ .

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

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