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Generalization of Quasi-modular Extensions

  • El Hassane FliouetEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 228)

Abstract

Let K/k be a purely inseparable extension of characteristic \(p>0\). Let lm(K/k) and um(K/k) be the smallest extensions \(k \longrightarrow lm(K\)/\(k) \longrightarrow K\longrightarrow um(K\)/k) such that K/lm(K/k) and um(K/k)/k are modular. In this note, we continue to study the locus problem of lm(K/k) and um(K/k) relative to K/k. Thus improving ([3], Theorem 1.4), we show that lm(K/k) is nontrivial when K/k is of finite size, more precisely if K/k has a finite size and unbounded exponent, the same is true of K/lm(K/k). However, if K/k is of unbounded size, it may well be that we lose this property by obtaining lm(K/\(k)=K \). In the following, we will say that K/k is lq-modular (respectively, uq-modular) if lm(K/k)/k (respectively, um(K/k)/K) has an exponent. The first study of these two concepts devoted to the extensions of finite size is in [4, 6, 7]. However, the object of the present work consists to generalize the results of finite size to any extension. In particular, we treat the stability questions of the lq-modularity and the uq-modularity relative to inclusion, intersection, and product. Furthermore, we are interested by the questions about existence of the smallest extensions which preserve these concepts in the ascendant or descendant sense, and also to the questions of existence of the maximal subextensions (closures).

Keywords

Purely inseparable q-finite modular extension Lq-modular extension Up-modular 

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Regional Center for the Professions of Education and TrainingAgadirMorocco

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