The resolution property for schemes and stacks

By B. Totaro. 2002

156 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e0461429f48660b74c63ebaf780ac9d89919817c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Jarod Alper
2. David Rydh
3. P. Stellari
4. Yukinobu Toda
5. Jack Hall
6. Hsian-Hua Tseng
7. H. Iritani
8. Max Lieblich

In Mathematical Physics

1. P. Stellari
2. Hsian-Hua Tseng
3. Jarod Alper
4. Max Lieblich
5. D. Huybrechts
6. D. Freed
7. Jack Hall
8. David Rydh

Citations in the corpus

• Tensor categorical foundations of algebraic geometry (2014, result): “For further results we refer to [Tot04] and [Gro10].”
• On the geometry of Deligne-Mumford stacks (2009, background): “In particular, if scheme then so is P, and if X is quasi-projective then so is P. To see this, it suces to perform a base change to an ane scheme X0 ´etale over X and show that P◊X X0”
• Toric vector bundles, branched covers of fans, and the resolution property (2006, methodology): “ported by a Graduate Research Fellowship from the NSF. 1 2 SAM PAYNE Totaro asks the following question in much greater generality, for a natural class of stacks that includes all algebraic varieties [Tot04], but it remains open and interesting in this special case. Totaro’s Question. Does every alg”
• Grothendieck Duality for Deligne-Mumford Stacks (2008, methodology): “omputation of f!Fto the computation of f!O Y. 9 Proposition 1.20. Let f: X→Ybe a separated finite-Tor dimensional morphism of noetherian algebraic stacks, assume that Yhas the resolution property (see [Tot04] for a comprehensive study of the problem), and both Xand Yhave affine diagonal. Let Fbe an obj”
• Loop groups and twisted K‐theory I (2007, result): “(iii) The result is curiously similar to Totaro’s characterization of smooth quotient stacks as the Artin stacks where coherent sheaves admit resolutions by vector bundles [32].”
• Moduli of twisted sheaves (2004, methodology): “Recall that X has the resolution property if every coherent sheaf on X is the quotient of a locally free sheaf [58].”
• Algebraization and Tannaka duality (2014, background): “ every finitely presented quasi-coherent sheaf is the cokernel of a map of vector bundles; any scheme that is quasi-projective over an affine has this property, and this property is studied in depth in [Tot04]. The object FunL ⊗(Vect(X),Vect(S)) denotes the category of right exact symmetric monoidal f”
• Variation of geometric invariant theory quotients and derived categories (2012, background): “ X. In general, the requirement that, for any coherent sheaf, C, on an algebraic stack, one can find a finite-rank locally-free sheaf, E, and an epimorphism, E→C, is called the resolution property. See [Tot04] for a fuller discussion of the generality of this condition. 2.3. Background on local cohomo”
• Tensor generators on schemes and stacks (2013, methodology): “r some family of vector bundles. This fundamental property of the category of quasicoherent sheaves was studied by various authors before and still poses a challenging problem. We refer the reader to [Tot04] for a detailed history and its connection to K-theory. Here, we prove that for every quasico”
• Operational K-Theory (2013, background): “r quasiprojective, where coherent sheaves are not known to have resolutions by vector bundles. On such spaces, it is not known whether Grothendieck rings of vector bundles and perfect complexes agree [To2]. For instance, there are complete, singular, nonprojective toric threefolds, such as [Pa2, Exa”
• Equivalences of derived catgories of sheaves on smooth stacks (2004, background): “This gives an alternative proof of Totaro’s theorem ([ 14 ]) for our stack.”
• Noetherian approximation of algebraic spaces and stacks (2009, background): “ where V is an algebraic space. Remark (2.2). Relation with the resolution property — B. Totaro has shown that a normal noetherian stack is a basic stack if and only if it has the resolution property [Tot04]. By recent work of P. Gross, this also holds for non-normal noetherian stacks [Gro10, Thm. 6”
• A tour about existence and uniqueness of dg enhancements and lifts (2016, background): “Actually, in [66] the existence of enough locally free sheaves under these assumptions is left as an open question.”