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The higher direct image complex of a coherent sheaf (or finite complex ofcoherent sheaves) under a projective morphism is a fundamental constructionthat can be defined via a Cech complex or an injective resolution, bothinherently infinite constructions. Using exterior algebras and relativeversions of theorems of Beilinson and Bernstein-Gel'fand-Gel'fand, we give analternate description in finite terms. Using this description we cancharacterize the generic complex, over the variety of finite free complexes ofa given shape, as the direct image of an easily-described vector bundle. We canalso give explicit descriptions of the loci in the base spaces of flat familiesof sheaves in which some cohomological conditions are satisfied--for example,the loci where vector bundles on projective space split in a certain way, orthe loci where a projective morphism has higher dimensional fibers. Ourapproach is so explicit that it yields an algorithm suited for computer algebrasystems.  This new version uses relative projectives to give smaller direct imagecomplexes in many cases. It also includes many small clarifications.

Relative Beilinson monad and direct image for families of coherent sheaves