Wild ramification in the imperfect residue field case. (English) Zbl 0649.12011

Hyodo, Osamu

Galois representations and arithmetic algebraic geometry, Proc. Symp., Kyoto/Jap. 1985 and Tokyo/Jap. 1986, Adv. Stud. Pure Math. 12, 287-314 (1987).

The author studies wild ramification of complete discrete valuation fields without the assumption that the residue field is perfect.
Let \(K\) be a complete discrete valuation field of mixed characteristics \((0,p)\). Let \(H^q(G,A)\) be the \(q\)-th continuous cohomology group of a topological group \(G\) with coefficients in a topological \(G\)-module \(A\). Let \(G_E = \mathrm{Gal}(E_{\text{sep}}/E)\) be the absolute Galois group of a field \(E\), and let \((r)\) denote the \(r\)-th Tate twist for \(r\in\mathbb{Z}\). The author proves that if the residue field \(\bar K\) of \(K\) is separably closed, then the inflation map induces an isomorphism \(H^1(G_k,\mathbb{Z}_p(r)) \overset{\simeq}{\rightarrow}H^1(G_K,\mathbb{Z}_p(r))\) for a suitable defined “canonical subfield” \(k\) of \(K\), \(r\ne 1\). A result of Miki for \(r=0\) is thus extended.
Let \(e_k\) be the absolute ramification index of \(K\) and \(L \mid K\) a cyclic extension. Miki showed that if \(e_k<p-1\), then \(\bar L\mid \bar K\) is separable; and if \([L:K]=[\bar L:\bar K]\), then the inseparable degree \([\bar L:\bar K]_{\text{insep}}\leq p^{e_k}\). The author gives a more precise estimate depending only on \(e_k\). He proves that \([\bar L:\bar K]_{\text{insep}}\leq p^{f(e_k)}\) and that if \([L:K]=[\bar L:\bar K]\), then \([\bar L:\bar K]_{\text{insep}}\leq p^{g(e_k)}\) where \(f\) and \(g\) are certain integer valued functions. The paper contains other interesting results.
[For the entire collection see Zbl 0632.00004.]

Reviewer: John N. Mordeson (Omaha)

MSC:

11S15	Ramification and extension theory
11S25	Galois cohomology
12F15	Inseparable field extensions

Keywords:

Galois cohomology groups; local class field theory; wild ramification of complete discrete valuation fields; inseparable degree

Citations:

Zbl 0632.00004