## 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].

### MSC:

 11S20 Galois theory 11S15 Ramification and extension theory

### Citations:

Zbl 0853.11096; Zbl 0873.11063