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We establish an equivalence between the stable category of coherent sheaves(satisfying a mild restriction) on a projective space and the homotopy categoryof a certain class of minimal complexes of free modules over the exterioralgebra Koszul dual to the homogeneous coordinate algebra of the projectivespace. We also relate these complexes to the Tate resolutions of the respectivesheaves. In this way, we extend from vector bundles to coherent sheaves theresults of Coand\u{a} and Trautmann [Trans. AMS 385 (2005)], which interpret interms 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 BGGcorrespondences for graded modules and for coherent sheaves and of the theoremof Eisenbud, Floystad and Schreyer [Trans. AMS 355 (2003)] describing thelinear part of the Tate resolution associated to a coherent sheaf. Moreover, weprovide an explicit description of the quotient of the Tate resolution by itslinear strand corresponding to the module of global sections of the varioustwists of the sheaf.

The Horrocks correspondence for coherent sheaves on projective spaces