  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.
