Describing Coherent Sheaves on Projective Spaces via Koszul Duality

It is well known that there is a close connection between coherent sheaves on a projective space P(V ) where V is a a vector space over a field k, and finitely generated graded modules over the symmetric algebra S(V ). The Bernstein-Gelfand-Gelfand (BGG) correspondence [5] from 1978 relates coherent sheaves on P(V ) with graded modules over the exterior algebra Λ(V ∗) = ⊕Λi(V ∗) where V ∗ is the dual vector space of V . This correspondence may be seen as a composition of the first connection and the correspondence between (complexes of) graded modules over S(V ) and (complexes of) graded modules over Λ(V ∗) coming from the fact that S(V ) and Λ(V ∗) are dual Koszul algebras. This latter correspondence, called Koszul duality, stems again from [5], and is treated subsequently in [3] and [11]. Of course the relationship between coherent sheaves on P(V ) and graded modules over the symmetric algebra S(V ) has been widely used. In this paper we shall investigate in detail the BGG-correspondence between complexes of coherent sheaves and complexes of graded modules over the exterior algebra. Our claim is that for algebraic purposes, complexes of graded modules over the exterior algebra Λ(V ∗) may be a more natural tool for investigating complexes of coherent sheaves on P(V ) than are complexes of graded modules over the symmetric algebra S(V ). To mention some applications of this we give a strikingly simple algebraic construction of the Horrocks-Mumford bundle on P4, we get a very natural proof of the Castelnuovo-Mumford theorem [24] on the regularity of coherent sheaves and we also give a generalization of a theorem of Barth [1] on stable rank two sheaves on P2, to a form which holds for all coherent sheaves on a projective space P(V ) (and from which Barth’s theorem is an immediate corollary).