[Paper Review] Tilting theory of preprojective algebras and $c$-sortable elements

Yuta Kimura|arXiv (Cornell University)|May 16, 2014

Algebraic structures and combinatorial models19 references3 citations

TL;DR

This paper establishes a derived equivalence between the graded stable category of submodules of a preprojective algebra factor Πw and the bounded derived category of a relative stable Auslander algebra, under the condition that w is c-sortable. It proves that Sub^Z Πw admits a tilting object M, and the endomorphism algebra of M is isomorphic to the stable Auslander algebra of a torsion-free class in mod kQ, with global dimension at most two.

ABSTRACT

For a finite acyclic quiver $Q$ and the corresponding preprojective algebra $Π$, we study the factor algebra $Π_w$ associated with a element $w$ in the Coxeter group introduced by Buan-Iyama-Reiten-Scott. The algebra $Π_w$ has a natural $\mathbb{Z}$-grading, and we prove that $\underline{\mathsf{Sub}}^{\mathbb{Z}}Π_w$ has a tilting object $M$. Moreover, we show that the endomorphism algebra of $M$ is isomorphic to the stable Auslander algebra of a certain torsion free class of $\mathsf{mod}\,kQ$.

Motivation & Objective

• To construct a derived category version of the Amiot-Reiten-Todorov equivalence between Sub Πw and cluster categories.
• To study the graded structure of the algebra Πw associated with a c-sortable element w in the Coxeter group of a finite acyclic quiver Q.
• To establish the existence of a tilting object in the graded stable category Sub^Z Πw when w is c-sortable.
• To describe the endomorphism algebra of the tilting object as a relative stable Auslander algebra and bound its global dimension.
• To provide a derived equivalence between Sub^Z Πw and the bounded derived category of the endomorphism algebra.

Proposed method

• Introduce a Z-grading on the preprojective algebra factor Πw using the orientation of the quiver Q.
• Define c-sortable elements as those w ∈ W admitting a reduced expression w = c(0)⋯c(m) with nested supports and subwords of the Coxeter element c.
• Construct a tilting object M = ⊕_{i=0}^m (Π_{c(0)⋯c(i)})(i) in Sub^Z Πw for c-sortable w.
• Use the theory of cotilting modules and relative stable categories to analyze the endomorphism algebra Aw = End^Z_Πw(M).
• Prove that Aw is isomorphic to End_kQ(M₀)/[T], where M₀ is the degree-zero part of M and T is a tilting kQ-module.
• Apply cotilting theory and long exact sequences in Hom^T to bound the global dimension of Aw by 2.

Experimental results

Research questions

• RQ1 Does the graded stable category Sub^Z Πw admit a tilting object when w is c-sortable?
• RQ2 What is the structure of the endomorphism algebra of the tilting object in Sub^Z Πw?
• RQ3 Can a derived equivalence be established between Sub^Z Πw and the bounded derived category of the endomorphism algebra?
• RQ4 What is the global dimension of the endomorphism algebra of the tilting object?
• RQ5 How does the endomorphism algebra relate to the stable Auslander algebra of a torsion-free class in mod kQ?

Key findings

• For any c-sortable element w in the Coxeter group of a finite acyclic quiver Q, the category Sub^Z Πw admits a tilting object M = ⊕_{i=0}^m (Π_{c(0)⋯c(i)})(i).
• The endomorphism algebra Aw = End^Z_Πw(M) is isomorphic to End_kQ(M₀)/[T], where M₀ is the degree-zero part of M and T is a tilting kQ-module such that Sub T has M₀ as an additive generator.
• The global dimension of Aw is at most two, as shown via cotilting theory and the vanishing of higher Ext groups.
• There exists a triangle equivalence Sub^Z Πw ≃ D^b(Aw), establishing a derived equivalence between the graded stable category and the bounded derived category of the endomorphism algebra.
• The construction provides a derived version of the Amiot-Reiten-Todorov equivalence, linking the category Sub Πw to cluster categories via graded structures.
• The global dimension bound of two for Aw follows from a general result on relative stable Auslander algebras, using the n-step cotilting resolution process with n=2.