Donovan’s Conjecture

Donovan's conjecture is a finiteness principle for block theory: fix a prime $p$, an algebraically closed field $k$ of characteristic $p$, and a finite $p$-group $D$; then among the blocks of all group algebras $kG$ with defect group isomorphic to $D$, only finitely many Morita-equivalence classes occur. The conjecture is uniformly attributed to Donovan in the 1970s, although no uniquely identifiable first printed source appears to exist [Donovan1976]. It would mean that a defect group controls the possible module categories of its blocks up to a finite list.

Substantial cases are known. Külshammer's analysis of crossed products treats blocks with normal defect groups [Kuelshammer1990Donovan], and there has been significant progress for blocks with abelian defect groups [EatonLivesey2018Donovan], alongside verifications for various small and low-rank defect groups.

For an arbitrary fixed defect group the finiteness statement is unresolved. The conjecture remains open: a proof must bound the Morita types arising from all finite groups at once, while a counterexample would require an infinite family of pairwise non-Morita-equivalent blocks sharing a single defect group.