  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.
