Donovan's conjecture over fields and complete mixed-characteristic DVRs. Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.
released 2026-09-24 | 4 theorems · 52 lemmas · 71 proofs · 50,334 words |
PLAY LEVEL 1 »(pdf)
We prove Donovan's conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.
released 2026-09-25 | 4 theorems · 6 lemmas · 17 proofs · 11,242 words |
PLAY LEVEL 2 »(pdf)
We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over $W(\overline{\mathbb F}_p)$. The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.