Cartan型自同构群的幺幂根中的极值导长度

中文摘要 AI 辅助

在每个群$G^{(j)}\subseteq\gamma_{2^j}(G)$中，我们展示了一个自然族，其中每一步都达到等号。设$k$是特征$p\ge 5$的完美域，$U_S$是高度一特殊代数$S(n;1)^{(1)}$（$n\ge 3$）的自同构群的幺幂根，使得$\mathrm{Aut}(S(n;1)^{(1)})_{\mathrm{red}}^{\circ}\simeq U_S\rtimes\mathrm{GL}_n$。记$D=n(p-1)$，我们证明对所有$j$有$U_S^{(j)}=\gamma_{2^j}(U_S)$，因此$U_S$的幂零类为$D-2$，且其导长度达到该类所允许的最大值，即$\lceil\log_2(D-1)\rceil$。同样的结论对$U_S(\mathbf{F}_q)$成立，其Frattini商和下$p$-中心列以闭式确定。其机制是一个同余滤过，其分次李代数在有限多个次数之外具有精确的括号生成，例外由一个Cartier商控制，该商既出现在交换化中，也出现在五项同调序列中。非光滑自同构概形包含$U_S$的一个典范Frobenius加厚，其Verschiebung滤过和分布代数被计算。在任意除幂高度上，de Rham复形上的Cartier-Verschiebung态射诱导出受限导子上的显式映射。在高度一中，我们确定了秩至少为二的Witt族和Hamiltonian族的两个序列，并将接触自同构群概形与坐标接触稳定化子等同。对于Witt根，同余滤过本身就是下中心列，且$\mathrm{cl}(U_W)=D-1$，$\mathrm{dl}(U_W)=\lceil\log_2 D\rceil$。对于Hamiltonian根，它由通量特征和socle线截出；设$D_H=2r(p-1)$，$r\ge 2$，则有$\mathrm{cl}(U_H)=D_H-3$，$\mathrm{dl}(U_H)=\lceil\log_2(D_H-2)\rceil$。因此，三者都达到了其类所允许的最大导长度。

英文摘要

Over perfect fields of characteristic $p\ge5$, the unipotent radicals of the height-one special, Witt-Jacobson, and Hamiltonian automorphism groups in at least three, two, and four variables, respectively, attain the largest derived length allowed by their nilpotency classes. More strongly, every term of each derived series equals the corresponding power-of-two term of the lower central series. These equalities hold on finite-field points, with explicit Frattini quotients and lower $p$-central series. For the special family, a Cartier quotient measures the difference between the congruence and lower central filtrations. The Hamiltonian correction comes from flux and the socle line; the Witt-Jacobson family has none. We also identify the contact automorphism group scheme and, at arbitrary divided-power height, compute the Cartier-Verschiebung maps on restricted derivations and the Verschiebung filtration and distribution algebra of the Frobenius thickening of the special radical.

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43 pages