[Paper Review] A Note on Auslander Bounds

Jiaqun Wei|arXiv (Cornell University)|Feb 8, 2008

Algebraic structures and combinatorial models16 references5 citations

TL;DR

This paper establishes that for a module M over an artin algebra R, if Ext^i_R(M, M ⊕ R) vanishes for all sufficiently large i, then the projective dimension of M equals lAb|modRM—the minimal bound on Ext vanishing across all modules in modR. This result supports the Auslander-Reiten Conjecture and implies that if lAb|modRM is finite for all M, then the Wakamatsu-tilting and Auslander-Reiten Conjectures hold for R.

ABSTRACT

Let R be a ring and modR be the category of all left R-modules possessing of finitely generated projective resolutions. Let lAb|modRM denote the minimal nonnegative integer m (or ∞ if no such integer exists) such that Ext i R (M, N) = 0 for all i> m and all N ∈ modR satisfying Ext i R (M, N) = 0 for all sufficiently large i. Auslander once conjectured lAb|modRM < ∞ for every M ∈ modR for any artin algebra R, but it is now known that the Auslander Conjecture fails by a counterexample due to Jorgensen and Sega. We show that if M ∈ modR and Ext i R (M, M ⊕ R) = 0 for all sufficiently large i, then the projective dimension of M coincides with lAb|modRM. The result is clearly relational to the Auslander-Reitein Conjecture which states that that if M ∈ modR and Ext i R(M, M ⊕ R) = 0 for all i> 0 then M is projective. As corollaries, we obtain that if R satisfies the condition that lAb|modRM < ∞ for every M ∈ modR, then an equivalent version of the Wakamatsu-tilting Conjecture, particularly the Auslander-Reitein Conjecture, holds for R, i.e., an R-module T ∈ modR is tilting if and only if (1) Ext i R(T, T) = 0 for all i> 0 and (2) there is an exact sequence 0 → R → T0 · · · → Tn → 0 for some n, with each Ti ∈ addT. We also investigate the Gorenstein Symmetry Conjecture and the finitistic Auslander conjecture which is stronger than the finitistic dimension conjecture and give some partial answers and consequences.

Motivation & Objective

• To investigate the relationship between the vanishing of Ext^i_R(M, M ⊕ R) for large i and the finiteness of lAb|modRM.
• To determine conditions under which lAb|modRM is finite and how this relates to the Auslander-Reiten Conjecture.
• To explore implications of finite lAb|modRM for the Wakamatsu-tilting Conjecture and related homological conjectures.
• To provide partial answers to the Gorenstein Symmetry Conjecture and the finitistic Auslander Conjecture.

Proposed method

• Define lAb|modRM as the minimal nonnegative integer m such that Ext^i_R(M, N) = 0 for all i > m and all N ∈ modR with Ext^i_R(M, N) = 0 for large i.
• Use the condition Ext^i_R(M, M ⊕ R) = 0 for all sufficiently large i to prove that pd(M) = lAb|modRM.
• Apply the theory of tilting modules and exact sequences in modR to analyze the equivalence between tilting conditions and vanishing Ext groups.
• Establish implications for the Wakamatsu-tilting Conjecture by showing that finite lAb|modRM implies the conjecture holds for R.
• Use homological algebra techniques, including projective resolutions and Ext vanishing, to derive consequences for the Gorenstein Symmetry Conjecture.
• Investigate the finitistic Auslander Conjecture as a stronger variant of the finitistic dimension conjecture, using the lAb|modRM framework.

Experimental results

Research questions

• RQ1 Under what conditions does the projective dimension of a module M equal lAb|modRM?
• RQ2 How does the vanishing of Ext^i_R(M, M ⊕ R) for large i relate to the finiteness of lAb|modRM?
• RQ3 What are the implications of lAb|modRM < ∞ for all M ∈ modR on the Wakamatsu-tilting and Auslander-Reiten Conjectures?
• RQ4 Can the Gorenstein Symmetry Conjecture be partially resolved using the lAb|modRM framework?
• RQ5 How does the finitistic Auslander Conjecture relate to the finiteness of lAb|modRM across all modules?

Key findings

• If Ext^i_R(M, M ⊕ R) = 0 for all sufficiently large i, then the projective dimension of M equals lAb|modRM.
• The finiteness of lAb|modRM for all M ∈ modR implies that the Wakamatsu-tilting Conjecture holds for R.
• Under the same finiteness condition, the Auslander-Reiten Conjecture is equivalent to the standard characterization of tilting modules via vanishing Ext and finite projective resolutions.
• The paper provides partial support for the Gorenstein Symmetry Conjecture through the lAb|modRM framework.
• The finitistic Auslander Conjecture is shown to be related to the uniform finiteness of lAb|modRM across all modules in modR.
• The result offers a new homological criterion for module projectivity and tilting behavior.