Resolutions of ideals associated to subspace arrangements

Abstract

Given a collection of $t$ subspaces in an $n$-dimensional $\mathbb{K} $-vector space $W$ we can associate to them $t$ vanishing ideals in the symmetric algebra $\mathcal{S}(W^*) = \mathbb{K}[x_1,x_2,\dots,x_n]$. As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal. 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 $t$ and they also showed that similar results hold for a more general class of ideals constructed from linear ideals. In this paper we show that analogous results hold when we replace the symmetric algebra $\mathcal{S}(W^*)$ with the exterior algebra $ \bigwedge(W^*)$ and work over a field of characteristic 0. To prove these results we rely on the functoriality of free resolutions and construct a functor $\Omega$ from the category of polynomial functors to itself. The functor $\Omega$ transforms resolutions of ideals in the symmetric algebra to resolutions of ideals in the exterior algebra.