Corollary

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

Algebra, chapter X, § 4. Produits de torsion

Let I be a right filtered preordered set, $(M_i, u_{ji})$ (resp. $(N_i, v_{ji})$ ) an inductive system of right $A$ -modules (resp. left) relative to I. The homomorphism of graded $k$ -modules

\lim_{\longrightarrow i \in I} \operatorname{Tor}^A(M_i, N_i) \to \operatorname{Tor}^A\left(\lim_{\longrightarrow i \in I} M_i, \lim_{\longrightarrow i \in I} N_i\right),

deduced from the canonical $A$ -homomorphisms $M_i \to \lim_{\longrightarrow} M_i$ and $N_j \to \lim_{\longrightarrow} N_j$ is bijective.

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

Cited by

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Cites

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