Corollary

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

Let $A$ be a ring, $M$ an $A$ -module, $(m_i)_{i \in I}$ a system of generators of $M$ and $a$ , the annihilator of $m_i$ . Then

\operatorname{Supp}(M) = \bigcup_{i \in I} V(a_i).

\operatorname{Supp}(M) = \bigcup_{i \in I} \operatorname{Supp}(Am_i)

by Proposition 16 (ii). On the other hand,

Am_i

is isomorphic to the

A

-module

A/a_i

and we have seen that

\operatorname{Supp}(A/a_i) = V(a_i)

(Example above).

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

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