Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conic modules, secondary fans and non-commutative resolutions

Aimeric Malter

Abstract

Faber, Muller and Smith used complete sums of conic modules to construct non-commutative crepant resolutions (NCCR) of simplicial toric algebras. We link these conic modules to the Bondal-Thomsen collection of line bundles on smooth toric DM stacks. This viewpoint allows us to establish computational results relating to conic modules, reducing the complexity of the combinatorics involved significantly. We formulate necessary and sufficient conditions for an incomplete sum of conic modules to give an NC(C)R of a toric algebra. Furthermore, we prove that to check if a toric algebra $R$ admits an NCCR in the form of $\End_R(\mathbb{B})$ for an incomplete sum of conic modules, we may reduce to a case where the class group of the affine toric variety $\spec R$ does not have torsion and verify the statement there. Finally, we treat the case of almost simplicial Gorenstein cones, i.e. cones with $|σ(1)|=\dim σ+1$, classifying when such cones admit NCCRs via endomorphism algebras of conic modules.

Conic modules, secondary fans and non-commutative resolutions

Abstract

Faber, Muller and Smith used complete sums of conic modules to construct non-commutative crepant resolutions (NCCR) of simplicial toric algebras. We link these conic modules to the Bondal-Thomsen collection of line bundles on smooth toric DM stacks. This viewpoint allows us to establish computational results relating to conic modules, reducing the complexity of the combinatorics involved significantly. We formulate necessary and sufficient conditions for an incomplete sum of conic modules to give an NC(C)R of a toric algebra. Furthermore, we prove that to check if a toric algebra admits an NCCR in the form of for an incomplete sum of conic modules, we may reduce to a case where the class group of the affine toric variety does not have torsion and verify the statement there. Finally, we treat the case of almost simplicial Gorenstein cones, i.e. cones with , classifying when such cones admit NCCRs via endomorphism algebras of conic modules.
Paper Structure (13 sections, 66 theorems, 109 equations, 3 figures, 1 table)

This paper contains 13 sections, 66 theorems, 109 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Structure and notation
  3. Acknowledgments
  4. Background
  5. NCCRs
  6. Conic modules
  7. The Bondal-Thomsen Collection
  8. Studying conic modules via the Bondal-Thomsen collection
  9. Substitution of conic modules and NCCRs
  10. Computing complexes of conic modules
  11. Complexes of conic modules and paths in the secondary fan
  12. Gorenstein cones
  13. Almost simplicial Gorenstein cones

Key Result

Theorem 1.3

Let $\sigma$ be a cone with associated toric algebra $R$ and collection of conic modules $\{A_v\}_{v\in S}$. For a subset $I$ of $S$, consider the incomplete direct sum of conic modules $\mathbb{B}=\bigoplus_{A_v\in S}A_v$. Then the endomorphism algebra $\Lambda'=\operatorname{End}_R(\mathbb{B})$ is

Figures (3)

  • Figure 1: Secondary fan and Zonotope for Example \ref{['Exa:Hexagon']}
  • Figure 2: The zonotope and interior lattice points
  • Figure 3: Valid paths for 3 dimensional almost simplicial Gorenstein cones.

Theorems & Definitions (147)

  • Conjecture 1.1
  • Theorem 1.3: = Theorem \ref{['Thm:NCRiffLock']}
  • Proposition 1.4: = Proposition \ref{['Prop:CplxViaPaths2']}
  • Theorem 1.5: = Theorem \ref{['Thm:Incr definitions agree']}
  • Theorem 1.6: = Theorem \ref{['Thm:AlmSimplConicNCCR']}
  • Definition 2.1
  • Definition 2.2
  • Conjecture 2.3
  • Proposition 2.4: Proposition 3.3 in SVdBtoricII
  • Theorem 2.5: = Theorem 3.12 in MS25
  • ...and 137 more