On the Bernstein-Gel’fand-Gel’fand correspondence and a result of Eisenbud, Fløystad, and Schreyer

Abstract

We show that a combination between a remark of I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand [2] and the idea, systematically investigated by D. Eisenbud, G. Fløystad and F.-O. Schreyer [3], of taking Tate resolutions over exterior algebras leads to quick proofs of the main results of [2] and [3] (Theorems 7 and 10 below). This combination is expressed by Lemma 6 from the text, a result which can be seen as a formula for computing hyperext groups on projective spaces in terms of linear algebra. We prove it directly, using only the cohomology of invertible sheaves on a projective space and a few basic facts about derived categories.