Lemma 51.5.1 (0EEZ)—The Stacks project

Loading web-font TeX/Math/Italic

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

$H^0_ T(M) = M$ , and
$\text{Supp}(M) \subset T$ .

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. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment