The resolution property for schemes and stacks

Abstract

Abstract. We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme.

Cone theorem via Deligne–Mumford stacks

“…Applying the arguments of Theorem 2.10 of [Ko92], we find that f * Ω Y has projective dimension 1. By the results of [To02], there is a global resolution of f * Ω Y by locally free sheaves,…”

Orbifold quantum Riemann–Roch, Lefschetz and Serre

“…Recall that a stack has the resolution property if every coherent sheaf is a quotient of a vector bundle (see for instance Totaro [51]). …”

On Coverings of Deligne–mumford Stacks and Surjectivity of the Brauer Map

“…In any discussion of quotient stacks, it is worth drawing a comparison with a related condition for algebraic stacks, known as the resolution property, which asserts that every coherent sheaf admits a surjection from a locally free coherent sheaf. The resolution property is discussed in some detail in a recent paper by Totaro [15]. For the stacks in Theorem 2.2, it is known (cf.…”