  The purpose of this paper is to verify a conjecture of Gross under mild
hypothesis: all reduced, separated, and excellent schemes have the resolution
property away from a closed subset of codimension at least three. Our technique
uses formal-local descent and the existence of affine flat neighborhoods to
reduce the problem to constructing certain modules over commutative rings. Once
in the category of modules we exhibit enough locally free sheaves directly,
thereby establishing the resolution property for a specific class of algebraic
spaces. A crucial step is showing it suffices to resolve a single coherent
sheaf.
