Lemma 48.22.3 (0AWK)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.22: Dualizing modules
Lemma 48.22.3 (cite)

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Lemma 48.22.3. Let $X$ be a connected Noetherian scheme and let $\omega _ X$ be a dualizing module on $X$ . The support of $\omega _ X$ is the union of the irreducible components of maximal dimension with respect to any dimension function and $\omega _ X$ is a coherent $\mathcal{O}_ X$ -module having property $(S_2)$ .

Proof. By our conventions discussed above there exists a dualizing complex $\omega _ X^\bullet$ such that $\omega _ X$ is the leftmost nonvanishing cohomology sheaf. Since $X$ is connected, any two dimension functions differ by a constant (Topology, Lemma 5.20.3). Hence we may use the dimension function associated to $\omega _ X^\bullet$ (Lemma 48.2.7). With these remarks in place, the lemma now follows from Dualizing Complexes, Lemma 47.17.5 and the definitions (in particular Cohomology of Schemes, Definition 30.11.1). $\square$

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