Essential Dimension and Faithful Rank of Finite p-Gerbes
Let $p\neq\operatorname{char}(k)$. We extend the Karpenko--Merkurjev theorem from classifying stacks of finite $p$-groups to arbitrary finite gerbes whose geometric inertia groups are $p$-groups, without assuming that the gerbe is neutral or that its band is represented by a group scheme over the base field. We prove that the essential dimension at $p$ is exactly the minimum faithful rank obtained after prime-to-$p$ base change, equivalently the faithful rank over a $p$-closure. We also prove a relative form of the theorem for locally full morphisms of finite $p$-gerbes: the relative faithful rank equals the supremum of the essential $p$-dimensions of the fibers. Finally, we introduce the quotient compression dimension, defined using tame quotient singularities with prescribed fundamental gerbe. For every finite $p$-gerbe $\mathcal{G}/k$ we show that its prime local version satisfies $$ \mathrm{ed}_k(\mathcal{G};p) \leq \operatorname{qcdim}_p(\mathcal{G}) \leq \mathrm{ed}_k(\mathcal{G};p)+1. $$ Thus essential dimension at $p$ determines, up to at most one dimension, the smallest quotient singularity realizing the gerbe after prime-to-$p$ localization.