Ramification filtration of the Galois group of a local field. II. (English. Russian original) Zbl 0884.11047

Proc. Steklov Inst. Math. 208, 15-62 (1995); translation from Tr. Mat. Inst. Steklova 208, 18-69 (1995).
[For Part I, see V. A. Abrashkin, Tr. St-Peterbg. Mat. Obshch. 3, 47-127 (1995; Zbl 0853.11096), as well as the English translation in Proceedings of the St. Petersburg Math. Soc. 3, Transl., Ser. 2, Am. Math. Soc. 166, 35-100 (1995; Zbl 0873.11063).]
Let \(K\) be a complete local field of characteristic \(p>0\) with a residue field \(k\). The Galois group \(Gal(K^{sep}/K)=\Gamma\) has a ramification filtration \(\{\Gamma^v\}_{v\in\mathbb{Q}_+}\). Let \(I\) be the union of all \(\Gamma^v\) with \(v>0\). \(I\) is a free pro-\(p\)-group. The paper contains an explicit description of the image of the ramification filtration in the group \(I/C_p\), where \(C_p=\{\text{all commutators of order}>p-1\}\). As a first step the author introduces a non-commutative generalization of the Artin-Schreier theory for \(p\)-extensions of nilpotency class \(<p\). It works for arbitrary fields of characteristic \(p>0\) and is grounded on an equivalence of category of finite Lie \(\mathbb{Z}_p\)-algebras \(L\) (of nilpotency class \(<p\)) and of category of finite \(p\)-groups \(G\) with the same condition (\(G:=G(L)\)). This theory has an independent interest. Next, the author identifies \(I/C_p\) with \(G({\mathcal L})\) for \({\mathcal L}\) a free Lie pro-\(\mathbb{Z}_p\)-algebra. He assumes that \(k=\overline\mathbb{F}_p\). Then an explicit filtration \({\mathcal L}^{(v)}\) in \({\mathcal L}\) is defined and it is shown that it coincides with the ramification filtration under the constructed identification. The proofs are given modulo 3-commutators and the general case will be considered in the next paper. Some version of the main theorem for \(k=\mathbb{F}_p\) and the Galois group of the maximal \(p\)-extension are also given.
For the entire collection see [Zbl 0863.00012].


11S20 Galois theory
11S15 Ramification and extension theory