The resolution property of algebraic surfaces

TL;DR

This paper proves that on separated algebraic surfaces, every coherent sheaf can be expressed as a quotient of a locally free sheaf, extending to certain 2-dimensional schemes over noetherian rings.

Contribution

It establishes the resolution property for separated algebraic surfaces and extends the result to 2-dimensional schemes proper over noetherian rings.

Findings

Every coherent sheaf on such surfaces is a quotient of a locally free sheaf.

Includes schemes that are not normal, reduced, quasiprojective, or embeddable into toric varieties.

Methods apply to arbitrary 2-dimensional schemes proper over a noetherian ring.

Abstract

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods extend to arbitrary $2$-dimensional schemes that are proper over a noetherian ring.