Definition 100.12.2. 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 94.16.5) 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
\[ \dim _ x(\mathcal{X}) = \dim _ u(U)-\dim _{e(u)}(R_ u). \]
with notation as in Lemma 100.12.1.
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