Donovan’s conjecture, blocks with abelian defect groups and discrete valuation rings

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 O . Consequences are that Donovan’s conjecture holds for 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 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 O -blocks is a consequence of conjectures predicting bounds on the O -Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field