Euler–Koszul homology in algebra and geometry

Homological methods are combined with the machinery of D-modules to study multivariate hypergeometric systems and monomial ideals. It is known that A-hypergeometric systems have constant rank for generic parameters. As a consequence of our main result, we produce a combinatorial formula for their rank at any parameter. Our methods induce a geometric stratification of the parameter space that refines its stratification by rank and yield a bound on rank through a homogenization process. A byproduct of our homological methods is a simpler proof for the classification of A-hypergeometric systems up to D-isomorphism. We also derive an explicit formula for the rank of a generalized A-hypergeometric system of monomial type. Finally, using hypergeometric techniques we generalize Reisner\u27s classical criterion: the exponent complexes of an arbitrary monomial ideal topologically detect its Cohen–Macaulay property