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Non-commutative crepant resolutions for (almost) simplicial toric algebras

Aimeric Malter, Artan Sheshmani

Abstract

Given a rational convex polyhedral Gorenstein cone constructed as cone over a lattice polytope P, we establish that toric non-commutative crepant resolutions (NCCRs) of its associated toric algebra descend to toric NCCRs of the algebras associated to faces of the polytope P. As consequence, we present two new, short proofs to the existence of toric NCCRs for simplicial affine toric Gorenstein algebras and for almost simplicial affine toric Gorenstein algebras, i.e. those associated to cones $σ$ with $\dimσ+1$ extremal rays.

Non-commutative crepant resolutions for (almost) simplicial toric algebras

Abstract

Given a rational convex polyhedral Gorenstein cone constructed as cone over a lattice polytope P, we establish that toric non-commutative crepant resolutions (NCCRs) of its associated toric algebra descend to toric NCCRs of the algebras associated to faces of the polytope P. As consequence, we present two new, short proofs to the existence of toric NCCRs for simplicial affine toric Gorenstein algebras and for almost simplicial affine toric Gorenstein algebras, i.e. those associated to cones with extremal rays.
Paper Structure (8 sections, 16 theorems, 24 equations, 2 figures)

This paper contains 8 sections, 16 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure and Notation
  3. Acknowledgements
  4. Toric geometry and non-commutative resolutions
  5. Cox stacks and cohomology computations
  6. Non-commutative crepant resolutions
  7. Descending NCCRs
  8. Application to (almost) simplicial toric algebras

Key Result

Theorem 1

(Theorem Thm:FaceOfReflexiveTORICNCCR) Let $Q\subset \mathbb{R}^k$ be a lattice polytope and let $\sigma$ be the cone $\sigma=\operatorname{Cone}(Q\times\{1\})\subset \mathbb{R}^{k+1}$ together with its toric algebra $R_\sigma=k[\sigma^\vee\cap M_\sigma]$, where $M_\sigma$ is the character lattice o

Figures (2)

  • Figure 1: Example of a polytope $Q$ with fan $\Sigma$ for $P=\operatorname{Conv}(v_1,v_2,v_3,v_4)$.
  • Figure 2: A drawing of $\Delta$ and $2\Delta$.

Theorems & Definitions (32)

  • Theorem
  • Corollary
  • Theorem
  • Definition 2.1
  • Theorem 2.2: Theorem 4.12 in FK18
  • Proposition 2.3
  • Definition 2.4
  • Corollary 2.5
  • Definition 2.6
  • Lemma 2.7: Lemma 4.4 in Efimov14
  • ...and 22 more