Corollary

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

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

Let $M$ be a finitely generated $A$ -module and $n$ its annihilator. For every ideal $a$ of $A$ , $\operatorname{Supp}(M/aM) = V(a) \cap V(n) = V(a + n)$ .

$M/aM = M \otimes_A (A/a)$ and $A/a$ is finitely generated.

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

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