发表机构
Scuola Normale Superiore（比萨高等师范学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将Karpenko–Merkurjev定理推广至几何惯性群为p-群的有限p-挠子，明确了其本质维数与忠实秩的关系，引入商压缩维数并建立了其与本质p-维数的界。
AI 中文摘要
设p≠char(k)，我们将Karpenko–Merkurjev定理从有限p-群的分类栈推广到几何惯性群为p-群的任意有限挠子，不要求挠子是中性的或其band由基域上的群概形表示。我们证明p处的本质维数恰为素数至p基变换后得到的最小忠实秩，等价于p闭包上的忠实秩；还证明了有限p-挠子的局部满态射的定理的相对形式：相对忠实秩等于纤维的本质p-维数的上确界。最后，我们引入商压缩维数，用具有指定基本挠子的 tame 商奇点定义，对每个有限p-挠子G/k，证明其素局部版本满足：ed_k(G;p) ≤ qcdim_p(G) ≤ ed_k(G;p)+1，因此p处的本质维数在最多差一个维数的范围内确定了素数至p局部化后实现该挠子的最小商奇点。
英文摘要
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.
