The resolution property holds away from codimension three

Abstract

The purpose of this paper is to verify a conjecture of Gross under mild hypotheses: all reduced, separated, and excellent schemes have the resolution property away from a closed subset of codimension ≥ 3 \geq 3 . 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.