Free resolutions for polynomial functors

Given a homogeneous functor L (e.g., L = ∧^k or L = ⊗^k and a finite free resolution F of a module M over a commutative ring with unit R, we construct in a canonical way a finite free complex C_L (F) that approximates a resolution of L (M), and whose acyclicity properties do not depend on the characteristic of R. We provide a criterion for the acyclicity of C_L (F). As an application we prove in general the conjecture of Buchsbaum and Eisenbud on the structure of the lower order minors of the differentials in a finite free resolution (previously known only for ℚ-algebras).