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On two examples by Iyama and Yoshino

Bernhard Keller, Daniel Murfet, Michel Van den Bergh

TL;DR

The paper studies two Iyama–Yoshino isolated singularity examples and gives two new routes to the stable category of maximal Cohen–Macaulay modules. One route uses cluster-tilting theory to realize the rigid CM category as an orbit category controlled by a square root of the AR-translate, yielding CY-3 structures and connections to generalized Kronecker quivers; the other leverages Orlov’s graded singularity framework to relate graded MCM categories to derived categories of coherent sheaves on Proj A and their semi-orthogonal decompositions, with explicit realizations for Veronese subrings. It derives explicit identifications underline{MCM}(R) ≅ D^b(mod(kQ_n)) / (\tau^{1/2}[-1]) in the two main examples (with n = 3,6) and clarifies how the grading shift interacts with Serre duality and the Gorenstein parameter a. A key side result shows that, for Gorenstein local rings with isolated singularities, completion induces an equivalence up to direct summands between singularity categories, extending to graded contexts via Orlov’s results. Overall, the work provides two complementary, tractable frameworks for understanding singularity categories in isolated, graded settings and enhances the bridge between representation theory and algebraic geometry.

Abstract

In the recent paper "Mutation in triangulated categories and rigid Cohen-Macaulay modules" Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov's result on the graded singularity category. We obtain some new results on the singularity category of isolated singularities which may be interesting in their own right.

On two examples by Iyama and Yoshino

TL;DR

The paper studies two Iyama–Yoshino isolated singularity examples and gives two new routes to the stable category of maximal Cohen–Macaulay modules. One route uses cluster-tilting theory to realize the rigid CM category as an orbit category controlled by a square root of the AR-translate, yielding CY-3 structures and connections to generalized Kronecker quivers; the other leverages Orlov’s graded singularity framework to relate graded MCM categories to derived categories of coherent sheaves on Proj A and their semi-orthogonal decompositions, with explicit realizations for Veronese subrings. It derives explicit identifications underline{MCM}(R) ≅ D^b(mod(kQ_n)) / (\tau^{1/2}[-1]) in the two main examples (with n = 3,6) and clarifies how the grading shift interacts with Serre duality and the Gorenstein parameter a. A key side result shows that, for Gorenstein local rings with isolated singularities, completion induces an equivalence up to direct summands between singularity categories, extending to graded contexts via Orlov’s results. Overall, the work provides two complementary, tractable frameworks for understanding singularity categories in isolated, graded settings and enhances the bridge between representation theory and algebraic geometry.

Abstract

In the recent paper "Mutation in triangulated categories and rigid Cohen-Macaulay modules" Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov's result on the graded singularity category. We obtain some new results on the singularity category of isolated singularities which may be interesting in their own right.

Paper Structure

This paper contains 18 sections, 23 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Notations and conventions
  4. First approach to the second example
  5. Some preliminaries on tilting complexes
  6. A square root of $\tau$ for a generalized Kronecker quiver
  7. A $3$-Calabi-Yau category with a $3$-cluster tilting object
  8. Proof of Theorem \ref{['ref-1.3-2']}
  9. The singularity category of graded Gorenstein rings
  10. Orlov's results
  11. The action of the shift functor on the singularity category
  12. The Serre functor for a graded Gorenstein ring
  13. The Gorenstein parameter of a Veronese subring
  14. The Iyama-Yoshino examples (again)
  15. Example \ref{['ref-1.1-0']}
  16. ...and 3 more sections

Key Result

Theorem 1.3

KeRe Assume that ${\mathcal{T}}$ is $k$-linear algebraic Krull-Schmidt $2$-Calabi-Yau category with a cluster tilting object $T$ such that $A=\operatorname {End}(T)$ is hereditary. Then there is an exact equivalence betweem ${\mathcal{T}}$ and the orbit category $D^b(\operatorname{mod}(A))/(\tau[-1]

Theorems & Definitions (50)

  • Example 1.1
  • Example 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Lemma 4.2.1
  • proof
  • Lemma 4.2.2
  • proof
  • ...and 40 more