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The Stacks project

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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