Lemma 51.5.1. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. For an $A$-module $M$ the following are equivalent

1. $H^0_ T(M) = M$, and

2. $\text{Supp}(M) \subset T$.

The category of such $A$-modules is a Serre subcategory of the category $A$-modules closed under direct sums.

Proof. The equivalence holds because the support of an element of $M$ is contained in the support of $M$ and conversely the support of $M$ is the union of the supports of its elements. The category of these modules is a Serre subcategory (Homology, Definition 12.10.1) of $\text{Mod}_ A$ by Algebra, Lemma 10.40.9. We omit the proof of the statement on direct sums. $\square$

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