Universal Picard-Vessiot Rings and Galois Groups

Abstract

Let K denote any differential field such that its field of constants C={a∈K|a′=0} is algebraically closed, has characteristic 0 and is different from K. The neutral tannakian category Diff K of differential modules over K is equivalent to the category Repr H of all finite dimensional representations (over C) of some affine group scheme H over C (see Appendices B.2 and B.3 for the definition and properties). Let C be a full subcategory of Diff K that is closed under all operations of linear algebra, i.e., kernels, cokernels, direct sums, and tensor products. Then C is also a neutral tannakian category and equivalent to Repr G for some affine group scheme G.