Sheaf cohomology and free resolutions over exterior algebras

Abstract

We derive an explicit version of the Bernstein-Gel’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 >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 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.