Proposition 7

Tag 0D4G, permanent. Cite it as [Bourbaki, Tag 0D4G].

Commutative Algebra, chapter X, § 3. Profondeur et dimension homologique

Let $A$ be a ring, $x$ a cancellable element of $A$ , $M$ an $A$ -module such that the homothety $x_M$ is injective, $N$ an $A$ -module annihilated by $x$ , $n$ an integer. The canonical homomorphisms of $(A/xA)$ -modules

\mathrm{Ext}^n_{A/xA}(M/xM, N) \longrightarrow \mathrm{Ext}^n_A(M, N),

\mathrm{Ext}^n_A(N, M) \longrightarrow \mathrm{Ext}^{n-1}_{A/xA}(N, M/xM),

\mathrm{Tor}^A_n(M, N) \longrightarrow \mathrm{Tor}^{A/xA}_n(M/xM, N)

deduced from the exact sequences (4), (5) and (6) are isomorphisms.

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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