Abstract
We develop Auslander's theory of coherent functors in the case of functors on modules of finite type over a noetherian ringA. In particular, the duality of coherent functors, which interchanges representable functors and tensor products, plays a special role. We apply these coherent functors to study cohomology of a flat family of sheaves on projective space over an affine base schemeT=SpecA. These results form a basic tool which is used in forthcoming work on the variation of the Rao module in a flat family of curves in P3.
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