  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.
