Théorème de la clôture $lq$-modulaire et applications
Volume 122 / 2011
Abstract
Let $K$ be a purely inseparable extension of a field $k$ of characteristic $p\not=0$. Suppose that $[k:k^{p}]$ is finite. We recall that $K/k$ is $lq$-modular if $K$ is modular over a finite extension of $k$. Moreover, there exists a smallest extension $m/k$ (resp. $M/K$) such that $K/m$ (resp. $M/k$) is $lq$-modular. Our main result states the existence of a greatest $lq$-modular and relatively perfect subextension of $K/k$. Other results can be summarized in the following:
1. The product of $lq$-modular extensions over $k$ is $lq$-modular over $k$.
2. If we augment the ground field of an $lq$-modular extension, the $lq$-modularity is preserved. Generally, for all intermediate fields $K_1$ and $K_2$ of $K/k$ such that $K_1/k$ is $lq$-modular over $k$, $K_1(K_2)/K_2$ is $lq$-modular.
By successive application of the theorem on $lq$-modular closure (our main result), we deduce that the smallest extension $m/k$ of $K/k$ such that $K/m$ is $lq$-modular is non-trivial (i.e. $m\not = K$). More precisely if $K/k$ is infinite, then $K/m$ is infinite.
