Resolutions of monomial ideals and cohomology over exterior algebras

This paper studies the homology of finite modules over the exterior algebra $E$ of a vector space $V$. To such a module $M$ we associate an algebraic set $V_E(M)\subseteq V$, consisting of those $v\in V$ that have a non-minimal annihilator in $M$. A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a `depth formula'. Explicit results are obtained for $M=E/J$, when $J$ is generated by products of elements of a basis $e_1,\dots,e_n$ of $V$. A (infinite) minimal free resolution of $E/J$ is constructed from a (finite) minimal resolution of $S/I$, where $I$ is the squarefree monomial ideal generated by `the same' products of the variables in the polynomial ring $S=K[x_1,\dots,x_n]$. It is proved that $V_E(E/J)$ is the union of the coordinate subspaces of $V$, spanned by subsets of $\{\,e_1,\dots,e_n\,\}$ determined by the Betti numbers of $S/I$ over $S$.