Lemma 107.5.9 (0DRN)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 107: The Geometry of Algebraic Stacks
Section 107.5: Dimension theory of algebraic stacks
Lemma 107.5.9 (cite)

Lemma 107.5.9. Suppose given a Cartesian square of morphisms of locally Noetherian stacks

$\xymatrix{ \mathcal{T}' \ar[d]\ar[r] & \mathcal{T} \ar[d] \\ \mathcal{X}' \ar[r] & \mathcal{X} }$

in which the vertical morphisms are locally of finite type. If $t' \in |\mathcal{T}'|$ , with images $t$ , $x'$ , and $x$ in $|\mathcal{T}|$ , $|\mathcal{X}'|$ , and $|\mathcal{X}|$ respectively, then $\dim _{t'}(\mathcal{T}'_{x'}) = \dim _{t}(\mathcal{T}_ x).$

Proof. Both sides can (by definition) be computed as the dimension of the same fibre product. $\square$

The Stacks project

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