The resolution property holds away from codimension three

TL;DR

This paper proves that reduced, separated, and excellent schemes possess the resolution property outside codimension three, using formal-local descent and module construction techniques to extend known results.

Contribution

It verifies Gross's conjecture for a broad class of schemes by demonstrating the resolution property away from codimension three, employing novel descent and module methods.

Findings

Resolution property holds away from codimension three for certain schemes.

Reduction to module construction simplifies the proof.

Establishment of the resolution property for specific algebraic spaces.

Abstract

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.