Lemma 48.22.4. Let X/A with \omega _ X^\bullet and \omega _ X be as in Example 48.22.1. Then
H^ i(\omega _ X^\bullet ) \not= 0 \Rightarrow i \in \{ -\dim (X), \ldots , 0\} ,
the dimension of the support of H^ i(\omega _ X^\bullet ) is at most -i,
\text{Supp}(\omega _ X) is the union of the components of dimension \dim (X), and
\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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