Horrocks theory and the Bernstein-Gelfand-Gelfand correspondence

Abstract

We construct an explicit equivalence between a category of complexes over the exterior algebra, which we call HT-complexes, and the stable category of vector bundles on the corresponding projective space, and establish a relation between HT-complexes and the Tate resolutions over the exterior algebra, which had been described by D. Eisenbud, G. Floystad, F.O. Schreyer in math.AG/0104203. The correspondence between HT-complexes and stable classes of vector bundles essentially translates into more fancy terms former results of Trautmann on representing Koszul complexes, which, in turn, were influenced by ideas of Horrocks. However, the relation between the Tate resolutions over the exterior algebra and HT-complexes might be new, although, perhaps, not a surprise to experts.