Proposition 5

Tag 0DY9, permanent. Cite it as [Bourbaki, Tag 0DY9].

Algebra, chapter X, § 6. Utilisation de résolutions non canoniques

If A is left noetherian and if M is a finitely generated A-module, the canonical homomorphism (8) is bijective.

For let (X, p. 53, Prop. 6) $p : L \to M$ be a free resolution of M such that $L_n$ is of finite type for each $n$ . The canonical morphism of $k$ -complexes

u : \lim \mathrm{Homgr}_A(L, N_i) \to \mathrm{Homgr}_A(L, \lim N_i)

is bijective, hence so also is the homomorphism

\lim \mathrm{H}(\mathrm{Homgr}_A(L, N_i)) \to \mathrm{H}(\mathrm{Homgr}_A(L, \lim N_i))

deduced from the homomorphisms $\mathrm{H}(\mathrm{Homgr}(l, u_i))$ (X, p. 28, Prop. 1). One then concludes by Prop. 2 (X, p. 103) and Th. 1 (X, p. 100).

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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