Hypergeometric D-modules and twisted Gauss-Manin systems

TL;DR

This paper explores the algebraic structure of A-hypergeometric systems using Euler-Koszul complexes, comparing homology with D-module images, and extends the theory to infinite toric modules and localizations.

Contribution

It generalizes previous results by providing a simpler proof and extending the Euler-Koszul functor to a broader class of modules.

Findings

Euler-Koszul homology aligns with D-module direct images.

Extended Euler-Koszul functor to infinite toric modules.

Described multigraded localizations of Euler-Koszul homology.

Abstract

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. and yields a simpler, more algebraic proof. In the process we extend the Euler-Koszul functor a category of infinite toric modules and describe multigraded localizations of Euler-Koszul homology.