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Adv. Math., 2013, Volume 238, Pages 322–411 (Mi admat10)  

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

Abstract: 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.

Funding Agency Grant Number
National Science Foundation DMS-0500762
DMS-0901373
CCF-0952591
Russian Foundation for Basic Research 11-01-00145-a
Ministry of Education and Science of the Russian Federation NSh-4713.2010.1
MK-4881.2011.1
11.G34.31.0023
City University of New York 60001-40 41
H. Gillet was supported by the grants NSF DMS-0500762 and DMS-0901373. S. Gorchinskiy was supported by the grants RFBR 11-01-00145-a, NSh-4713.2010.1, MK-4881.2011.1, and AG Laboratory GU-HSE, RF government grant, ag. 11 11.G34.31.0023. A. Ovchinnikov was supported by the grants: NSF CCF-0952591 and PSC-CUNY No. 60001-40 41.


DOI: https://doi.org/10.1016/j.aim.2013.02.006


Bibliographic databases:

Document Type: Article
MSC: Primary 12H05; Secondary 12H20; 13N10; 20G05; 20H20; 34M15
Received: 21.05.2012
Language: English

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