Relative Beilinson monad and direct image for families of coherent sheaves

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 Čech complex or an injective resolution, both inherently infinite constructions. Using free resolutions it can be defined in finite terms. Using exterior algebras and relative versions of theorems of Beilinson and Bernstein-Gel$'$fand-Gel$'$fand, we give an alternate and generally more efficient description in finite terms.

Using this exterior algebra description we can characterize the generic finite free complex of a given shape as the direct image of an easily-described vector bundle. We can also 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.