Exterior Algebra Resolutions Arising from Homogeneous Bundles

When R is a commutative ring, the minimal free resolution of a map Ra → Rb and symmetric and skew-symmetric maps Ra → Ra, under suitable generality conditions, are well known and have been developed by a series of authors. See [2, A2.6] for an overview. In this note we do an analog for the exterior algebra E = ⊕ ∧i V on a finite dimensional vector space V and general graded maps Ea → E(1)b, and general graded symmetric and skew-symmetric maps Ea → E(1)a. Since E is both a projective and injective E-module, by taking a free (projective) and cofree (injective) resolution of such maps, there is associated an unbounded acyclic complex of free E-modules, called a Tate resolution, see [3] or [4]. Via the Bernstein-Gel’fand-Gel’fand (BGG) correspondence, this corresponds to a complex of coherent sheaves on the projective space P(V ∗). We show that in all the cases above (with the dimension of V not too low), this complex actually reduces to a coherent sheaf. We describe these coherent sheaves and also describe completely the Tate resolutions. These descriptions turn out to be simpler to work out than in the corresponding commutative case, and, maybe at first surprising, the descriptions are also more geometric. In fact not only are we able to describe the Tate resolutions and coherent sheaves associated to the maps stated above, but, using the theory of representations of reductive groups, we are able to describe the Tate resolutions and coherent sheaves associated to vast larger classes of natural maps Ea → E(r)b, something which would have required considerably more effort for commutative rings. There is only one catch related to all our descriptions. We must assume that the dimension of V is not too small compared to a and b. For instance for a general map Ea → E(1)b we must assume that the dimension of V is ≥ a+ b − 1. In case the dimension of V is smaller than this, the nature of the problem changes, and we do not investigate this case.