Sheaf Cohomology and Free Resolutions over Exterior Algebras

Abstract

In this paper we study the Bernstein-Gel'fand-Gel'fand (BGG) correspondence linking sheaves on a projective space to graded modules over an exterior algebra. We give an explicit construction of a Beilinson monad for a sheaf on projective space. The explicitness allows us to to prove two conjectures about the morphisms in the monad. We also construct all the monads for a sheaf that can be built from sums of line bundles. A large subclass, containing many monads introduced by others, are uniquely characterized by simple numerical data. Our methods also yield an efficient method for machine computation of the cohomology of sheaves. Along the way we study minimal 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. This paper is a joint, extended version of papers previously written by Eisenbud-Schreyer and by Fl{\o}ystad seperately.