Parameterized Picard-Vessiot extensions and Atiyah extensions

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.

90 pages, minor corrections

References in corpus (17)

• The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem
• The notion of category over an algebraic stack
• Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations
• Isomonodromic differential equations and differential categories
• Zariski Closures of Reductive Linear Differential Algebraic Groups
• Differential Tannakian Categories
• Unipotent differential algebraic groups as parameterized differential Galois groups
• Tannakian approach to linear differential algebraic groups
• Monodromy groups of parameterized linear differential equations with regular singularities
• Generalised Hasse-Schmidt varieties and their jet spaces
• The inverse problem of differential Galois theory over the field R(z)
• Extensions of differential representations of SL(2) and tori
• Scalar Extension of Abelian and Tannakian Categories
• Splitting fields and general differential Galois theory
• Parameterized generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier
• On the Grothendieck conjecture on p-curvatures for q-difference equations
• Hypertranscendance et Groupes de Galois aux differences

Cited by in corpus (25)

• Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations
• Isomonodromic differential equations and differential categories
• Zariski Closures of Reductive Linear Differential Algebraic Groups
• Unipotent differential algebraic groups as parameterized differential Galois groups
• A density theorem for parameterized differential Galois theory
• A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function
• Extensions of differential representations of SL(2) and tori
• Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence
• Computing the differential Galois group of a parameterized second-order linear differential equation
• On parameterized differential Galois extensions
• Differential Galois Groups over Laurent Series Fields
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
• Difference integrability conditions for parameterized linear difference and differential equations
• Intrinsic approach to Galois theory of q-difference equations, with the preface to Part 4 "The Galois D-groupoid of a q-difference system'' by Anne Granier
• Parameterized Differential Equations over k((t))(x)
• Tannakian categories with semigroup actions
• Descent for differential Galois theory of difference equations. Confluence and q-dependency
• Real Liouvillian Extensions of Partial Differential Fields
• Triviality of differential Galois cohomologies of linear differential algebraic groups
• Real Picard-Vessiot theory
• Calculating Galois groups of third order linear differential equations with parameters
• Differential Galois Theory and Isomonodromic Deformations
• Picard--Vessiot structures
• Linear Algebraic Groups as Parameterized Picard-Vessiot Galois Groups
• Picard-Vessiot theory for real partial differential fields