Proposition 8

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

Algebra, chapter X, § 4. Produits de torsion

Let I (resp. J) be a right filtered preordered set, (( $M_\alpha$ , $u_{\alpha', \alpha}$ ) (resp. (( $N_\beta$ , $v_{\beta', \beta}$ ))) an inductive system of right A-modules (resp. left) relative to I (resp. J). The homomorphism of graded k-modules

\lim_{\longrightarrow (\alpha, \beta) \in I \times J} \mathrm{Tor}^A(M_\alpha, N_\beta) \to \mathrm{Tor}^A\left( \lim_{\longrightarrow \alpha \in I} M_\alpha, \lim_{\longrightarrow \beta \in J} N_\beta \right),

deduced from the canonical A-homomorphisms $M_\alpha \to \lim_{\longrightarrow} M_\alpha$ and $N_\beta \to \lim_{\longrightarrow} N_\beta$ , is bijective.

In particular, taking $J = I$ and remarking that the $(\alpha, \alpha), \alpha \in I$ , form a cofinal subset of $I \times I$ , one obtains :

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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