Lemma 48.21.2. Let S be a Noetherian scheme and let \omega _ S^\bullet be a dualizing complex. Let f : X \to S be of finite type and let \omega _ X^\bullet be the dualizing complex normalized relative to \omega _ S^\bullet . For all x \in X we have
where \delta _ S, resp. \delta _ X is the dimension function of \omega _ S^\bullet , resp. \omega _ X^\bullet , see Lemma 48.2.7.
Proof. We may replace X by an affine neighbourhood of x. Hence we may and do assume there is a compactification X \subset \overline{X} over S. Then we may replace X by \overline{X} and assume that X is proper over S. We may also assume X is connected by replacing X by the connected component of X containing x. Next, recall that both \delta _ X and the function x \mapsto \delta _ S(f(x)) + \text{trdeg}_{\kappa (f(x))}(\kappa (x)) are dimension functions on X, see Morphisms, Lemma 29.52.3 (and the fact that S is universally catenary by Lemma 48.2.7). By Topology, Lemma 5.20.3 we see that the difference is locally constant, hence constant as X is connected. Thus it suffices to prove equality in any point of X. By Properties, Lemma 28.5.9 the scheme X has a closed point x. Since X \to S is proper the image s of x is closed in S. Thus we may apply Lemma 48.21.1 to conclude. \square
Comments (0)