Definition 98.12.2. Let $\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$. Let $x \in |\mathcal{X}|$ be a point of $\mathcal{X}$. Let $[U/R] \to \mathcal{X}$ be a presentation (Algebraic Stacks, Definition 92.16.5) where $U$ is a scheme and let $u \in U$ be a point that maps to $x$. We define the *dimension of $\mathcal{X}$ at $x$* to be the element $\dim _ x(\mathcal{X}) \in \mathbf{Z} \cup \infty $ such that

with notation as in Lemma 98.12.1.

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