Adv. Math., 2013, Volume 238, Pages 322–411
Parameterized Picard–Vessiot extensions and Atiyah extensions
H. Gilleta, S. Gorchinskiyb, A. Ovchinnikovcd
a University of Illinois at Chicago, Department of Mathematics Statistics, and Computer Science, 851 S Morgan Street
b Steklov Mathematical Institute, Gubkina str. 8
c CUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd.
d CUNY Graduate Center, Department of Mathematics, 365 Fifth Avenue
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.
MSC: Primary 12H05; Secondary 12H20; 13N10; 20G05; 20H20; 34M15
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