Definition 107.4.3. Let $\mathcal{X}$ be an algebraic stack locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ is a morphism where $k$ is a field of finite type over $S$. The multiplicity of a formal branch of $\mathcal{X}$ through $x_0$ is the multiplicity of the corresponding irreducible component of $\mathop{\mathrm{Spec}}(A)$ for any choice of versal ring to $\mathcal{X}$ at $x_0$ (see discussion above).
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