Recent zbMATH articles in MSC 12J25https://www.zbmath.org/atom/cc/12J252021-04-16T16:22:00+00:00WerkzeugEliminating tame ramification generalizations of Abhyankar's lemma.https://www.zbmath.org/1456.120052021-04-16T16:22:00+00:00"Dutta, Arpan"https://www.zbmath.org/authors/?q=ai:dutta.arpan"Kuhlmann, Franz-Viktor"https://www.zbmath.org/authors/?q=ai:kuhlmann.franz-viktorBefore stating the theorems which are proved in this paper, we have to recall some properties of valued fields. For every valued field \((K,v)\), \(v\) extends to the algebraic closure \(K^{ac}\) of \(K\). Now \((K,v)\) is henselian if \(v\) extends in a unique way to every algebraic extension. Every valued field \((K,v)\) embeds in a minimal henselian valued field \((K^h,v)\), which is uniquely determined up to isomorphism and is called its Hensel closure. The extension \((K^h|K,v)\) is immediate, i.e. \(vK^h=vK\) and \(K^hv=Kv\), where \(vK\) denotes the valuation group of \((K,v)\) and \(Kv\) its residue field.
An algebraic extension \((L|K,v)\) of henselian valued fields is tame if, for every finite subextension \((E|K,v)\), the characteristic of \(Kv\) doesn't divide \((vE:vK)\), \(Ev|Kv\) is separable and \([E:K]=(vE:vK)[Ev:Kv]\). Next, we have the following sequence of valued fields:
\(K\subseteq K^h\subseteq K^i\subseteq K^r\subseteq K^s\subseteq K^{ac}\).
Here \(K^s\) is the separable closure of \(K\), \(K^r\) is called its ramification field and \(K^i\) its inertia field. These fields satisfy the following properties:
\(vK^i=vK\), \(K^iv=K^rv\), the extensions \(K^iv|Kv\), \(K^r|K\) and \(K^i|K\) are Galois,
\(\mathrm{Gal}(K^iv|Kv)\simeq \mathrm{Gal}(K^i|K^h)\), and if the characteristic of \(Kv\) is \(p>0\), then \(\mathrm{Gal}(K^s|K^r)\) is a pro-\(p\)-group, \(\mathrm{Gal}(K^r|K^i)\) doesn't contain any element of order \(p\) and \(K^sv|K^iv\) is purely inseparable.
In the following the extension \(L|K\) is algebraic, and \(F|K\) is an arbitrary extension.
First, assume that \(L\subseteq K^r\). The authors prove that \(LF\subseteq F^r\), \(v(LF)=vL+vF\). Further, \(LF\subseteq F^i\Leftrightarrow vL\subseteq vF\). Next, assume that \(L\subseteq K^i\). Then \(LF\subseteq F^i\), \((LF)v=Lv\cdot Fv\). Furthermore, \(LF\subseteq F^h\Leftrightarrow Lv\subseteq Fv\).
The authors deduce that if the rational rank of \(vK\) is \(1\), \((L\cdot K^h|K^h,v)\) is tame, \((vL:vK)\) and \((vF:vK)\) are finite, then \((v(LF):vL)\) is the lcm of \((vL:vK)\) and \((vF:vK)\). In particular, \(v(LF)=vF\) if, and only if, \((vL:vK)\) divides \((vF:vK)\) (\(\Leftarrow\) is an Abhyankar's lemma).
At the end of the paper they construct an example which shows that this result fails for rational rank \(2\).
In the next theorem, they assume that \(L|K\) is a normal extension and that the characteristic of \(Kv\) is \(p>0\). The greatest subgroup of \((vL)\) which contains \(vK\) and such that \(p\) doesn't divide the order of any non zero element modulo \(vK\) is denoted by \((vL)_{p'}\). First they prove that the quotient group \(v(LF)/((vL)_{p'}+vF)\) is a \(p\)-group, and they deduce that \(v(LF)/vF\) is a \(p\)-group if, and only if, \((vL)_{p'}\subseteq vF\). Next, they assume that \((vL)_{p'}=vK\) and they show that the maximal separable subextension of \((LF)v|(Lv)^sFv\) is a \(p\)-extension.
They give an example where the previous extension is nontrivial, although \(Lv=(Lv)^s=Kv\).
They also gives examples of several situations. We can quote a case where \(K=L^d=L^i\subsetneq L^r=L\), but \(F=(FL)^h\subsetneq (FL)^i=(FL)^r=FL\); so \(F=FL^i\subsetneq (FL)^i\) and \(F\subsetneq L^rF=(FL)^i\).
In another example we have \(K=L^h\subsetneq L^i=L^r=L\), \(F\subsetneq LF^h=(LF)^i=(LF)^r=LF\) (so \(F\subsetneq L^iF=(LF)^d=LF\)). They also investigate examples where \(F\) is a rational function field.
Reviewer: Gerard Leloup (Le Mans)