Parameterized Picard-Vessiot extensions and Atiyah extensions

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• 作者：Henri Gillet^a ; ^{gillet@uic.edu} ; Sergey Gorchinskiy^b ; ^{gorchins@mi.ras.ru} ; Alexey Ovchinnikov^c ; ^d ; ^{aovchinnikov@qc.cuny.edu}
• 关键词：primary ; 12H05 ; secondary ; 12H20 ; 13N10 ; 20G05 ; 20H20 ; 34M15
• 刊名：Advances in Mathematics
• 出版年：2013
• 出版时间：1 May, 2013
• 年：2013
• 卷：238
• 期：Complete
• 页码：322-411
• 全文大小：594 K

文摘

Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is to the Galois theory of linear differential equations with parameters. Namely, we show the existence of a parameterized Picard-Vessiot extension and, therefore, the Galois correspondence for many differential fields with, possibly, non-differentially closed fields of constants, that is, fields of functions of parameters. Other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic, which will appear in a separate paper. This application is based on differential categories developed in the present paper, and not just differential algebraic groups and their representations.