Sheaf Cohomology and Free Resolutions over Exterior Algebras

In this paper we derive an explicit version of the BernsteinGel’fand-Gel’fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its “Koszul dual” exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the “linear part” of a resolution over the exterior algebra. We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their “linear parts” in the sense that erasing all terms of degree > 1 in the complex yields a new complex which is eventually exact. As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data. Let V be a finite dimensional vector space over a field K, and let W = V ∗ be the dual space. In this paper we will study complexes and resolutions over the exterior algebra E = ∧V and their relation to modules over S = SymW and sheaves on projective space P(W ). In this paper we study the Bernstein-Gel’fand-Gel’fand (BGG) correspondence [1978], usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on P(W ) and the stable category of finitely generated graded modules over E. Its essential content is a functorR from complexes of graded S-modules to complexes of graded E-modules, and its adjoint L. For example, if M = ⊕iMi is a graded S-module (regarded as a complex with just one term) then ∗ The first and third authors are grateful to the NSF for partial support during the preparation of this paper. The third author wishes to thank MSRI for its hospitality. AMS Classification. Primary: 14F05, 14Q20, 16E05