The Horrocks correspondence for coherent sheaves on projective spaces

TL;DR

This paper extends the Horrocks correspondence to coherent sheaves on projective spaces, establishing an equivalence with complexes over the exterior algebra and relating it to Tate resolutions, thus generalizing previous results for vector bundles.

Contribution

It introduces a new equivalence between stable categories of coherent sheaves and homotopy categories of complexes, extending the Horrocks correspondence beyond vector bundles.

Findings

Established an equivalence between sheaves and complexes over the exterior algebra.

Connected Tate resolutions to the BGG correspondence for sheaves.

Provided explicit descriptions of Tate resolution quotients.

Abstract

We establish an equivalence between the stable category of coherent sheaves (satisfying a mild restriction) on a projective space and the homotopy category of a certain class of minimal complexes of free modules over the exterior algebra Koszul dual to the homogeneous coordinate algebra of the projective space. We also relate these complexes to the Tate resolutions of the respective sheaves. In this way, we extend from vector bundles to coherent sheaves the results of Coand\u{a} and Trautmann [Trans. AMS 385 (2005)], which interpret in terms of the BGG correspondence the results of Trautmann [Math. Ann. 237 (1978)] about the correspondence of Horrocks [Proc. London. Math. Soc. 14 (1964)], [Asterisque 71-72 (1980)]. We also give direct proofs of the BGG correspondences for graded modules and for coherent sheaves and of the theorem of Eisenbud, Floystad and Schreyer [Trans. AMS 355 (2003)] describing the linear part of the Tate resolution associated to a coherent sheaf. Moreover, we provide an explicit description of the quotient of the Tate resolution by its linear strand corresponding to the module of global sections of the various twists of the sheaf.