[Paper Review] Infinitely many algebras derived equivalent to a block

Jeremy Rickard|arXiv (Cornell University)|Oct 9, 2013

Algebraic structures and combinatorial models3 references3 citations

TL;DR

This paper demonstrates that certain block algebras of finite groups with abelian defect groups are derived equivalent to infinitely many non-Morita-equivalent algebras, using a construction based on syzygies of modules over symmetric algebras. The key result is that unbounded Cartan invariants in the derived equivalent algebras imply infinitely many Morita equivalence classes, challenging the finiteness assumptions in Donovan’s Conjecture.

ABSTRACT

We give a construction that in many cases gives a simple way to construct infinite families of algebras that are not Morita equivalent, but are all derived equivalent to the same block algebra of a finite group, and apply it to some small blocks. We make some remarks relating this construction to Donovan's Conjecture and Broue's Abelian Defect Group Conjecture.

Motivation & Objective

• To investigate whether derived equivalence to a block algebra implies finitely many Morita equivalence classes, particularly in relation to Donovan’s and Broué’s conjectures.
• To determine whether the existence of infinitely many derived equivalent algebras with unbounded Cartan invariants contradicts weaker forms of Donovan’s Conjecture.
• To provide a constructive method for generating infinite families of derived equivalent algebras that are not Morita equivalent.
• To explore whether such algebras can be ruled out as block algebras without relying on the Classification of Finite Simple Groups.

Proposed method

• Construct a tilting complex using projective covers and quotient modules indexed by a subset of simple modules.
• Use the duality in symmetric algebras to analyze Hom-spaces in the homotopy category of projective modules.
• Define a module $ M = \operatorname{Hom}_{A}(P_k, P_\epsilon) $ over the endomorphism algebra $ E $, and analyze its syzygies $ \Omega^s M $.
• Establish a recurrence relation $ a_{s+1} = 3a_s + 1 $ for the dimension of the socle of $ \Omega^s M $, showing exponential growth.
• Apply Theorem 2.3 to conclude that unbounded Cartan invariants imply infinitely many Morita equivalence classes of derived equivalent algebras.
• Use the structure of group algebras with abelian defect groups (e.g., $ D = C_3 \times C_3 $) to construct explicit examples with infinite derived equivalence classes.

Experimental results

Research questions

• RQ1 Are there infinite families of Morita equivalence classes of algebras derived equivalent to a given block algebra, even with bounded Cartan invariants?
• RQ2 Can the existence of infinitely many derived equivalent algebras with unbounded Cartan invariants be used to refute weaker forms of Donovan’s Conjecture?
• RQ3 Is it possible to prove that some derived equivalent algebras constructed via syzygy methods are not Morita equivalent to any block algebra without invoking the Classification of Finite Simple Groups?
• RQ4 Do derived equivalences preserve enough structural properties to distinguish block algebras from other derived equivalent algebras?

Key findings

• The syzygy dimensions $ a_s $ of the module $ M $ grow exponentially, satisfying the recurrence $ a_{s+1} = 3a_s + 1 $ with $ a_0 = a_1 = 2 $.
• The solution to the recurrence shows exponential growth: $ a_s = \left(1 - \frac{1}{\sqrt{5}}\right)\left(\frac{3+\sqrt{5}}{2}\right)^s + \left(1 + \frac{1}{\sqrt{5}}\right)\left(\frac{3-\sqrt{5}}{2}\right)^s $.
• Unbounded Cartan invariants in the derived equivalent algebras imply infinitely many Morita equivalence classes.
• The construction applies to block algebras of finite groups with abelian defect groups such as $ D = C_3 \times C_3 $, producing infinite families of derived equivalent but not Morita equivalent algebras.
• The method does not require the Classification of Finite Simple Groups, suggesting a potential path to distinguishing non-block algebras from derived equivalents.
• The result shows that Broué’s Abelian Defect Conjecture does not imply Donovan’s Conjecture in a straightforward way, as even bounded Cartan invariants may not suffice to ensure finitely many Morita classes.