Donovan\u2019s conjecture, blocks with abelian defect groups and discrete valuation rings

Abstract

We give a reduction to quasisimple groups for Donovan’s conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring {mathcal {O}}. Consequences are that Donovan’s conjecture holds for {mathcal {O}}-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan’s conjecture for {mathcal {O}}-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan’s conjecture for {mathcal {O}}-blocks is a consequence of conjectures predicting bounds on the {mathcal {O}}-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.