The Stacks project

Definition 107.5.7. If $f : \mathcal{T} \to \mathcal{X}$ is a locally of finite type morphism between locally Noetherian algebraic stacks, and if $t \in |\mathcal{T}|$ is a point with image $x \in |\mathcal{X}|$, then we define the relative dimension of $f$ at $t$, denoted $\dim _ t(\mathcal{T}_ x),$ 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 $t' \in |\mathcal{T} \times _{\mathcal{X}} \mathop{\mathrm{Spec}}k|$ mapping to $t$ under the projection to $|\mathcal{T}|$ (such a point $t'$ exists, by Properties of Stacks, Lemma 100.4.3; then

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

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