Lemma 10.39.9. Let $R$ be a ring and let $M$ be an $R$-module.

If $M$ is finite then the support of $M/IM$ is $\text{Supp}(M) \cap V(I)$.

If $N \subset M$, then $\text{Supp}(N) \subset \text{Supp}(M)$.

If $Q$ is a quotient module of $M$ then $\text{Supp}(Q) \subset \text{Supp}(M)$.

If $0 \to N \to M \to Q \to 0$ is a short exact sequence then $\text{Supp}(M) = \text{Supp}(Q) \cup \text{Supp}(N)$.

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