The Stacks project

Definition 106.5.2. If $f : T \to \mathcal{X}$ is a locally of finite type morphism from an algebraic space to an algebraic stack, and if $t \in |T|$ is a point with image $x \in | \mathcal{X}|$, then we define the relative dimension of $f$ at $t$, denoted $\dim _ t(T_ x),$ as follows: choose a morphism $\mathop{\mathrm{Spec}}k \to \mathcal{X}$, with source the spectrum of a field, which represents $x$, and choose a point $t' \in |T \times _{\mathcal{X}} \mathop{\mathrm{Spec}}k|$ mapping to $t$ under the projection to $|T|$ (such a point $t'$ exists, by Properties of Stacks, Lemma 99.4.3); then

\[ \dim _ t(T_ x) = \dim _{t'}(T \times _{\mathcal{X}} \mathop{\mathrm{Spec}}k ). \]

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