Proposition 18

Tag 04AG, permanent. Cite it as [Bourbaki, Tag 04AG].

Commutative Algebra, chapter II, § 4. Spectra of rings and supports of modules

Let $M, M'$ be two finitely generated modules over a ring $A$ ; then

\operatorname{Supp}(M \otimes_A M') = \operatorname{Supp}(M) \cap \operatorname{Supp}(M').

We need to prove that, if $p$ is a prime ideal of $A$ , the relations $(M \otimes_A M')_p \neq 0$ and " $M_p \neq 0$ and $M'_p \neq 0$ " are equivalent. As the $A_p$ -modules $M, \otimes_{A_p} M'_p$ and $(M \otimes_A M')_p$ are isomorphic (§ 2, no. 7, Proposition 18), our assertion follows from the following lemma:

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

Cited by

Nothing.

Cites

§ 2, no. 7, Proposition 18