Resolutions of monomial ideals and cohomology over exterior algebras

Abstract

This paper studies the homology of finite modules over the exterior algebra E E of a vector space V V . To such a module M M we associate an algebraic set V E ( M ) ⊆ V V_E(M)\subseteq V , consisting of those v ∈ V v\in V that have a non-minimal annihilator in M 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 M=E/J , when J J is generated by products of elements of a basis e 1 , … , e n e_1,\dots ,e_n of V V . A (infinite) minimal free resolution of E / J E/J is constructed from a (finite) minimal resolution of S / I S/I , where I I is the squarefree monomial ideal generated by ‘the same’ products of the variables in the polynomial ring S = K [ x 1 , … , x n ] S=K[x_1,\dots ,x_n] . It is proved that V E ( E / J ) V_E(E/J) is the union of the coordinate subspaces of V V , spanned by subsets of { e 1 , … , e n } \{\,e_1,\dots ,e_n\,\} determined by the Betti numbers of S / I S/I over S S .