Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Computing the higher direct image complex of coherent sheaves by Frank-Olaf SCHREYER, Universität des Saarlandes Abstract: (Joint work with David Eisenbud) The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Cech complex or an injective resolution, both inherently infinite constructions. Using exterior algebras and relative versions of theorems of Beilinson and Bernstein-Gel'fand-Gel'fand, we give an alternate description in finite terms. Using this description we can give explicit descriptions of the loci in the base spaces of flat families of sheaves in which some cohomological conditions are satisfied---for example, the loci where vector bundles on projective space split in a certain way, or the loci where a projective morphism has higher dimensional fibers. Our approach is so explicit that it yields an algorithm suited for computer algebra systems.