Resolutions of ideals of subspace arrangements

Abstract

Given a collection of t subspaces in an ndimensional vector space W we can associate to them t linear ideals in the symmetric algebra S(W ∗). Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of t linear ideals is at most t. In this paper we show that analogous results hold when we work over the exterior algebra ∧ (W ∗) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.