迭代联络的伽罗瓦论与非约伽罗瓦群
Galois theory for iterative connections and nonreduced Galois groups
摘要：
本文提出了一个具有迭代连接的模的理论。这个理论是一个推广的理论模块与连接的特点零模块环的任意特征。我们表明，这些模块与迭代连接（以及可积迭代连接的模块）形成一个Tannakian范畴，假设一些很好的性能的基础环，我们展示了如何推广到模块的计划。我们也将这些概念与A. Grothendieck为了将可积（普通）联络推广到有限特征。在光滑环上，我们得到了层化与可积迭代联络的等价性。进一步，在正特征的正则环上，我们证明了具有可积迭代联络的模范畴也等价于D.吉塞克在本文的第二部分，我们建立了一个Picard-Vessiot理论的解决方案领域。对于这样的Picard-Vessiot扩展，我们得到了一个Galois对应，它考虑到甚至非约闭子群计划的伽罗瓦群计划一方面和不可分的中间扩展的Picard-Vessiot扩展的另一方面。最后，我们比较我们的伽罗瓦论与伽罗瓦论的纯不可分的域扩张。
英文摘要：
This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these modules with iterative connection (and also the modules with integrable iterative connection) form a Tannakian category, assuming some nice properties for the underlying ring, and we show how this generalises to modules over schemes. We also relate these notions to stratifications on modules, as introduced by A. Grothendieck in order to extend integrable (ordinary) connections to finite characteristic. Over smooth rings, we obtain an equivalence of stratifications and integrable iterative connections. Furthermore, over a regular ring in positive characteristic, we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by D. Gieseker. In the second part of this article, we set up a Picard-Vessiot theory for fields of solutions. For such a Picard-Vessiot extension, we obtain a Galois correspondence, which takes into account even nonreduced closed subgroup schemes of the Galois group scheme on one hand and inseparable intermediate extensions of the Picard-Vessiot extension on the other hand. Finally, we compare our Galois theory with the Galois theory for purely inseparable field extensions.
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