The Stacks project

Lemma 48.22.4. Let $X/A$ with $\omega _ X^\bullet $ and $\omega _ X$ be as in Example 48.22.1. Then

  1. $H^ i(\omega _ X^\bullet ) \not= 0 \Rightarrow i \in \{ -\dim (X), \ldots , 0\} $,

  2. the dimension of the support of $H^ i(\omega _ X^\bullet )$ is at most $-i$,

  3. $\text{Supp}(\omega _ X)$ is the union of the components of dimension $\dim (X)$, and

  4. $\omega _ X$ has property $(S_2)$.

Proof. Let $\delta _ X$ and $\delta _ S$ be the dimension functions associated to $\omega _ X^\bullet $ and $\omega _ S^\bullet $ as in Lemma 48.21.2. As $X$ is proper over $A$, every closed subscheme of $X$ contains a closed point $x$ which maps to the closed point $s \in S$ and $\delta _ X(x) = \delta _ S(s) = 0$. Hence $\delta _ X(\xi ) = \dim (\overline{\{ \xi \} })$ for any point $\xi \in X$. Hence we can check each of the statements of the lemma by looking at what happens over $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ in which case the result follows from Dualizing Complexes, Lemmas 47.16.5 and 47.17.5. Some details omitted. The last two statements can also be deduced from Lemma 48.22.3. $\square$

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