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Algebra Seminar

Monday, February 26, 2018 - 3:15pm

Ray Hoobler



University of Pennsylvania

DRL 4N30

Let R be a differential algebra containing Q with a set D of commuting derivations, and let E be a free differential R module. We first show that there is a differential R algebra, CF, over which E has a basis of constants, F, satisfying the universal mapping property that if S is any differential R algebra over which E has a constant basis B, then there is a unique differential algebra homomorphism f:CF->S that sends the universal frame F to B. If C, the differential constants of R, is an algebraically closed field, we then easily construct a universal simple differential R algebra, the associated Picard-Vessiot extension, and show that it is a principal homogeneous space for an affine algebraic group defined over C. This leads naturally and directly to a Picard-Vessiot Galois theory. In particular, this theory applies to a smooth algebra R over the complex numbers. This approach also transfers immediately to parametrized Picard-Vessiot theory and difference Picard-Vessiot theory. Time permitting an extension to the case when C is not a field will also be discussed. Knowledge of Picard-Vessiot theory will not be assumed.