Ideals of Subspace Arrangements

Abstract

Given a collection of $t$ subspaces in an $n$-dimensional $mathbb{K} $-vector space $W$, we can associated 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 $t$. 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. We use our regularity bound on the intersection of $t$ linear ideals to prove Noether's degree bound on the minimal generating invariant polynomials of a finite group acting on $ bigwedge(W^*)$. We also provide a fast algorithm to compute the invariant monomials of a finite abelian group.