Hypergeometric D-modules and twisted Gauß–Manin systems

The Euler–Koszul complex is the fundamental tool in the homological study of A-hypergeometric differential systems and functions. We compare Euler–Koszul homology with D-module direct images from the torus to the base space through orbits in the corresponding toric variety. Our approach generalizes a result by Gel'fand et al. [I.M. Gel'fand, M.M. Kapranov, A.V. Zelevinsky, Generalized Euler integrals and A-hypergeometric functions, Adv. Math. 84 (2) (1990) 255–271, MR MR1080980 (92e:33015), Thm. 4.6] and yields a simpler, more algebraic proof.In the process we extend the Euler–Koszul functor to a category of infinite toric modules and describe multigraded localizations of Euler–Koszul homology.