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 VE(M) ⊆ V , consisting of those v ∈ 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 e1, . . . , en 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[x1, . . . , xn]. It is proved that VE(E/J) is the union of the coordinate subspaces of V , spanned by subsets of { e1, . . . , en } determined by the Betti numbers of S/I over S.