Coherent functors

We develop Auslander's theory of coherent functors in the case of functors on modules of finite type over a noetherian ring A. In particular, the dual ity of coherent functors, which interchanges representable functors and ten sor products, plays a special role. We apply these coherent functors to stu dy cohomology of a flat family of sheaves on projective space over an affin e base scheme T = Spec A. These results form a basic tool which is used in forthcoming work on the Variation of the Rao module in a flat family of cur ves in P-3. (C) 1998 Academic Press.