[Paper Review] Auslander-Gorenstein algebras, standardly stratified algebras and dominant dimensions

René Marczinzik|arXiv (Cornell University)|Oct 10, 2016

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper establishes a deep connection between Auslander-Gorenstein algebras, standardly stratified algebras, and dominant dimension by proving that for Auslander-Gorenstein algebras of Gorenstein dimension $ r $, the categories of modules with dominant dimension $ r-j $ and Gorenstein projective dimension $ j $ coincide. It further shows that when such algebras are properly stratified with a simple-preserving duality and the characteristic tilting module equals the characteristic cotilting module, the Gorenstein dimension is $ 2m $, and the categories of standardly filtered modules align precisely with dominant and Gorenstein projective categories.

ABSTRACT

We give new properties of algebras with finite Gorenstein dimension coinciding with the dominant dimension $\geq 2$, which are called Auslander-Gorenstein algebras in the recent work of Iyama and Solberg, see \cite{IyaSol}. In particular, when those algebras are standardly stratified, we give criteria when the category of (properly) (co)standardly filtered modules has a nice homological description using tools from the theory of dominant dimensions and Gorenstein homological algebra. We give some examples of standardly stratified algebras having dominant dimension equal to the Gorenstein dimension, including examples having an arbitrary natural number as Gorenstein dimension and blocks of finite representation-type of Schur algebras.

Motivation & Objective

• To explore the interplay between dominant dimension, Gorenstein homological algebra, and standardly stratified structures in finite-dimensional algebras.
• To identify conditions under which the categories of (co)standardly filtered modules admit a homological description via dominant and Gorenstein projective/injective dimensions.
• To classify representation-finite algebras in class $\mathcal{A}$ (quasi-hereditary with simple-preserving duality and dominant dimension ≥2), showing they are isomorphic to Auslander algebras of $K[x]/(x^3)$ or blocks of Schur algebras.
• To provide a homological characterization of the characteristic tilting module in special cases, linking it to projective and injective dimensions.
• To establish a formula for the relative Auslander-Reiten translate in the category $F(\overline{\Delta})$ for algebras satisfying the main theorem conditions.

Proposed method

• Define Auslander-Gorenstein algebras as finite-dimensional algebras where Gorenstein dimension $ r $ equals dominant dimension $ \geq 2 $, generalizing higher Auslander algebras.
• Use the equivalence $ Dom_{r-j} = GProj_j $ and $ Codom_{r-j} = GInj_j $ for $ j = 0,1,\dots,r $, linking dominant and Gorenstein homological dimensions.
• Apply Mueller’s theorem to compute dominant dimensions via $ Ext^i(M,M) $ vanishing up to a certain degree, using minimal projective resolutions.
• Construct explicit minimal projective resolutions of simple modules to verify $ Ext^i(S_n, S_n) = 0 $ for $ i = 1,\dots,2n-2 $, proving dominant dimension $ 2n-2 $.
• Use the generator-cogenerator $ A \oplus S_n $ and apply $ End_A(A \oplus S_n) $ to realize algebras $ B(n,1,\dots,1) $, showing they are higher Auslander algebras.
• Leverage the fact that symmetric algebras and generator-cogenerators ensure algebras lie in class $ \mathcal{A} $, enabling classification results.

Experimental results

Research questions

• RQ1 When does the category of (properly) (co)standardly filtered modules admit a homological description via dominant and Gorenstein projective/injective dimensions?
• RQ2 Under what conditions does the characteristic tilting module coincide with the characteristic cotilting module in Auslander-Gorenstein algebras?
• RQ3 Which representation-finite algebras in class $ \mathcal{A} $ exist, and what is their structure in terms of Auslander algebras or Schur algebra blocks?
• RQ4 What is the precise relationship between Gorenstein dimension, dominant dimension, and the structure of standardly stratified algebras in this setting?
• RQ5 How can the relative Auslander-Reiten translate be computed in $ F(\overline{\Delta}) $ for algebras satisfying the main duality and tilting conditions?

Key findings

• For an Auslander-Gorenstein algebra of Gorenstein dimension $ r $, the categories $ Dom_{r-j} $ and $ GProj_j $ coincide for all $ j = 0,1,\dots,r $, establishing a direct link between dominant and Gorenstein homological dimensions.
• When the algebra is properly stratified with a simple-preserving duality and the characteristic tilting module equals the characteristic cotilting module, the Gorenstein dimension is $ 2m $, where $ m $ is the projective dimension of the tilting module.
• In this case, $ F(\bar{\Delta}) = Dom_m = GProj_m $, $ F(\Delta) = Proj_m $, $ F(\bar{\nabla}) = Codom_m = GInj_m $, and $ F(\nabla) = Inj_m $, showing a complete homological equivalence.
• The algebra $ B(n,1,1,\dots,1) $ has dominant dimension $ 2n-2 $, global dimension $ 2n-2 $, and satisfies $ F(\Delta) = Dom_{n-1} = Proj_{n-1} $, confirming it is a higher Auslander algebra.
• All representation-finite algebras in class $ \mathcal{A} $ are isomorphic to the Auslander algebra of $ K[x]/(x^3) $ or to $ B(n,1,1,\dots,1) $, and are Morita equivalent to blocks of Schur algebras.
• The characteristic tilting module of $ B(n,1,1,\dots,1) $ is $ T = eA \oplus S_1 $, where $ eA $ is the minimal faithful projective-injective module, and $ \Omega^{n-1}((1-e)B) = S_1 $, confirming the tilting structure.