Parameterized Picard-Vessiot extensions and Atiyah extensions

Details Parameterized Picard-Vessiot extensions and Atiyah extensions Parameterized Picard-Vessiot extensions and Atiyah extensions

2011-10-16

arxiv.org/abs/1110.3526v2

Mathematics

Commutative Algebra

71 pages; Section 8.1 updated

Scientific paper

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 in 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.

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