[Paper Review] Cohen-Macaulay Auslander algebras of gentle algebras

Xinhong Chen, Ming Lu|arXiv (Cornell University)|Feb 13, 2015

Algebraic structures and combinatorial models38 references3 citations

TL;DR

This paper explicitly constructs the quiver and relations of the Cohen-Macaulay Auslander algebra for any gentle algebra Λ = KQ/I, proving that Λ is representation-finite if and only if its Cohen-Macaulay Auslander algebra is. It further shows that if the quiver Q has no loops and all Λ-modules are uniquely determined by their dimension vectors, then the same holds for the Auslander algebra's modules.

ABSTRACT

For any gentle algebra $Λ=KQ/\langle I angle$, following Kalck, we describe the quiver and the relations for its Cohen-Macaulay Auslander algebra $\mathrm{Aus}(\mathrm{Gproj}Λ)$ explicitly, and obtain some properties, such as $Λ$ is representation-finite if and only if $\mathrm{Aus}(\mathrm{Gproj}Λ)$ is; if $Q$ has no loop and any indecomposable $Λ$-module is uniquely determined by its dimension vector, then any indecomposable $\mathrm{Aus}(\mathrm{Gproj}Λ)$-module is uniquely determined by its dimension vector.

Motivation & Objective

• To explicitly describe the quiver and relations of the Cohen-Macaulay Auslander algebra Aus(Gproj Λ) for any gentle algebra Λ.
• To investigate the representation-finiteness of Λ in relation to its Cohen-Macaulay Auslander algebra.
• To determine conditions under which indecomposable modules over the Auslander algebra are uniquely determined by their dimension vectors.
• To explore structural properties of the Auslander algebra when Λ is schurian or has no loops.
• To establish connections between module uniqueness, quiver structure, and algebraic invariants in the context of gentle algebras.

Proposed method

• Construct the quiver QAus and ideal IAus of the Cohen-Macaulay Auslander algebra using the structure of Gorenstein projective modules over Λ.
• Apply known results on string modules and string algebras to classify indecomposable modules over the Auslander algebra.
• Use the fact that gentle algebras are CM-finite and Gorenstein to analyze the Auslander algebra's global dimension and module category.
• Leverage the string module parametrization of indecomposable modules over gentle algebras to transfer properties to the Auslander algebra.
• Employ dimension vector analysis and quiver combinatorics to study uniqueness conditions for modules.
• Use the structure of cycles and paths in the quiver to prove that strings in the Auslander algebra pass through vertices at most once, ensuring schurian properties.

Experimental results

Research questions

• RQ1 What is the explicit quiver and set of relations for the Cohen-Macaulay Auslander algebra of a gentle algebra?
• RQ2 When is the Cohen-Macaulay Auslander algebra of a gentle algebra representation-finite?
• RQ3 Under what conditions are indecomposable modules over the Cohen-Macaulay Auslander algebra uniquely determined by their dimension vectors?
• RQ4 How does the absence of loops in the quiver of a gentle algebra affect the uniqueness of modules in its Cohen-Macaulay Auslander algebra?
• RQ5 Is the Cohen-Macaulay Auslander algebra of a schurian gentle algebra also schurian and gentle?

Key findings

• The quiver and relations of the Cohen-Macaulay Auslander algebra Aus(Gproj Λ) are explicitly described in Theorem 3.5 for any gentle algebra Λ.
• Λ is representation-finite if and only if its Cohen-Macaulay Auslander algebra is representation-finite, as established in Theorem 4.4.
• If the quiver Q of Λ has no loops and every indecomposable Λ-module is uniquely determined by its dimension vector, then every indecomposable Aus(Gproj Λ)-module is also uniquely determined by its dimension vector, as shown in Theorem 4.6.
• If the Aus(Gproj Λ)-module category has two non-isomorphic indecomposables with the same dimension vector, then Λ must be isomorphic to K[X]/⟨X²⟩, as per Corollary 4.10.
• The Cohen-Macaulay Auslander algebra of a schurian gentle algebra is itself a schurian gentle algebra, as proven in Proposition 4.11.
• The proof of Proposition 4.11 relies on showing that any string module in the Auslander algebra passes through each vertex at most once, preserving the schurian property.