The Stacks project

Remark 105.5.6. For Deligne–Mumford stacks which are suitably decent (e.g. quasi-separated), it will again be the case that $\dim _ x(\mathcal{X})$ coincides with the topologically defined quantity $\dim _ x |\mathcal{X}|$. However, for more general Artin stacks, this will typically not be the case. For example, if $\mathcal{X} = [\mathbf{A}^1/\mathbf{G}_ m]$ (over some field, with the quotient being taken with respect to the usual multiplication action of $\mathbf{G}_ m$ on $\mathbf{A}^1$), then $|\mathcal{X}|$ has two points, one the specialisation of the other (corresponding to the two orbits of $\mathbf{G}_ m$ on $\mathbf{A}^1$), and hence is of dimension $1$ as a topological space; but $\dim _ x (\mathcal{X}) = 0$ for both points $x \in |\mathcal{X}|$. (An even more extreme example is given by the classifying space $[\mathop{\mathrm{Spec}}k/\mathbf{G}_ m]$, whose dimension at its unique point is equal to $-1$.)


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