Resumen de Purely inseparable extensions of rings
Purely inseparable extensions were originally defined in the setting of field extensions. They are an important object of interest in Commutative Algebra and have been studied extensively, in particular with a view to develop an analogue of the classical Galois theory established for separable field extensions. The notion of purely inseparable has been extended later to the setting of rings. Some authors have proposed different, but related, generalizations for this concept. This thesis studies purely inseparable extensions of rings following the perspective from by F. Pauer . His definition considers purely inseparable ring extensions as the smaller subclass of commutative ring extensions containing purely inseparable field extensions and that is closed by base change and faithfully flat descent. This thesis is focused on deepening the notion of purely inseparable ring extension and on generalizing some of the properties and results that are already known for purely inseparable field extensions to this broader context. In particular, we study purely inseparable extensions of rings from the perspective of differential operators and purely inseparable towers of rings using properties of projective modules. The thesis consists of four parts, which include the contents of the work in collaboration with D. Sulca and O. Villamayor. In the first part, we analyze the more general setting of finite extensions of rings A ⊆ C such that C is projective as an A-module. In particular, given such an extension we study some towers of rings A ⊆ B ⊆ C and we prove a generalization of the Jacobson-Bourbaki correspondence for finite field extensions to this class of extensions when Spec(A) and Spec(C) are homeomorphic. The second part begins with the study of purely inseparable extensions of rings in the exponent one case, which were introduced by S. Yuan as Galois extensions. We review the relation of these extensions with derivations, Kähler differentials, p-basis and differentiably simple rings. We provide a new characterization for Galois extensions by considering derivations as differential operators of order one. Moreover, we define the Yuan functor that parametrizes Galois towers of rings, we prove that it is representable by an scheme and we study its geometric properties. The third part is devoted to purely inseparable extensions of rings in general. We first review the original characterizations given by F. Pauer in terms of normal generating sequences and modules of differentials. Then we study these extensions from the perspective of differential operators and obtain the new characterization. We also use these techniques on differential operators to re-obtain a characterization involving the modules of principal parts. In addition, we study purely inseparable towers of rings and provide some sufficient conditions for a tower of rings A ⊆ B ⊆ C to be purely inseparable. The final part addresses modular extensions, which are a subclass of purely inseparable extensions. We extend the original definition of modular extensions of fields to the setting of rings using the property of being linearly disjoint. We prove that these extensions are exactly the purely inseparable ring extensions that satisfy certain conditions on linearly disjointedness
