Lemma 48.2.7. Let $X$ be a locally Noetherian scheme. Let $\omega _ X^\bullet $ be a dualizing complex on $X$. Then $X$ is universally catenary and the function $X \to \mathbf{Z}$ defined by

is a dimension function.

Lemma 48.2.7. Let $X$ be a locally Noetherian scheme. Let $\omega _ X^\bullet $ be a dualizing complex on $X$. Then $X$ is universally catenary and the function $X \to \mathbf{Z}$ defined by

\[ x \longmapsto \delta (x)\text{ such that } \omega _{X, x}^\bullet [-\delta (x)] \text{ is a normalized dualizing complex over } \mathcal{O}_{X, x} \]

is a dimension function.

**Proof.**
Immediate from the affine case Dualizing Complexes, Lemma 47.17.3 and the definitions.
$\square$

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