[Paper Review] A Homological dimension related to AB rings

Tokuji Araya|arXiv (Cornell University)|Apr 20, 2012

Algebraic structures and combinatorial models12 references3 citations

TL;DR

This paper introduces AB-dimension, a new homological invariant closely tied to AB rings—Gorenstein rings satisfying the Auslander condition. It establishes that AB-dimension satisfies a depth formula and fits between CI-dimension and G-dimension, with finite AB-dimension implying the Auslander-Reiten conjecture holds for such modules. The key contribution is proving that if a module M has finite AB-dimension and Ext^{>0}(M,M)=0, then M is free.

ABSTRACT

There are many homological dimensions which are closely related to ring theoretic properties. The notion of a AB ring has been introduced by Huneke and Jorgensen. It has nice homological properties. In this paper, we shall define a homological dimension which is closely related to a AB ring, and investigate its properties.

Motivation & Objective

• To define a new homological dimension, AB-dimension, that captures structural properties of AB rings.
• To investigate the relationship between AB-dimension and existing homological dimensions such as CI-dimension and G-dimension.
• To establish conditions under which AB-dimension is finite and to characterize AB rings via module-theoretic properties.
• To prove that modules of finite AB-dimension satisfying Ext^{>0}(M,M)=0 are free, supporting the Auslander-Reiten conjecture.

Proposed method

• AB-dimension is defined as the supremum of G-dimension and P_R(M), where P_R(M) = sup{P_R(M,N) | N ∈ M^⊥}.
• The paper uses minimal free resolutions and syzygy modules to analyze the behavior of Ext modules and define P_R(M,N).
• It applies Cohen-Macaulay approximations and properties of totally reflexive modules to relate AB-dimension to depth and G-dimension.
• Lemmas on Ext vanishing and depth formulas are used to prove that AB-dimension satisfies a depth formula: AB-dim_R(M) = depth R - depth M when finite.
• The proof of Theorem 1.3 uses contradiction: assuming M is not free leads to nonvanishing Ext^1(M,ΩM), contradicting P_R(M,ΩM)=0 under the given assumptions.
• The paper constructs a counterexample using a periodic module over a non-Cohen-Macaulay ring to show strict inequality AB-dim > G-dim can occur.

Experimental results

Research questions

• RQ1 When is AB-dimension finite for a given module over a commutative noetherian local ring?
• RQ2 How does AB-dimension relate to CI-dimension and G-dimension in the homological dimension chain?
• RQ3 What characterizes AB rings in terms of module classes closed under extension and finite AB-dimension?
• RQ4 Does the condition Ext^{>0}(M,M)=0 imply freeness for modules of finite AB-dimension?
• RQ5 Can AB-dimension be used to verify or extend the Auslander-Reiten conjecture?

Key findings

• AB-dimension satisfies the depth formula: if AB-dim_R(M) < ∞, then AB-dim_R(M) = depth R - depth M.
• For any module M, CI-dim_R(M) ≥ AB-dim_R(M) ≥ G-dim_R(M), establishing a new position in the homological dimension hierarchy.
• An R-module M is free if it has finite AB-dimension and Ext^{>0}_R(M,M)=0, confirming a special case of the Auslander-Reiten conjecture.
• There exist modules M with infinite AB-dimension but finite G-dimension, showing the AB-dimension is strictly stronger than G-dimension.
• When R is a complete intersection, it is an AB ring, and all modules have finite AB-dimension, generalizing known results about Gorenstein rings.
• The class of modules of finite AB-dimension is closed under extensions if and only if R is Gorenstein and AB-dimension is finite on all finite-length modules.