1 · The essentials
These manuscripts address whether an enormous collection of algebraic objects can fall into only finitely many types. The objects come from finite groups, and the rule for treating two objects as the same is called Morita equivalence. Each object has an associated group whose size supplies the bound in the question. The main manuscript claims that, once the underlying field is fixed and this size is bounded, only finitely many types occur. Its claim covers every prime characteristic, including , without requiring those associated groups to be abelian. A companion manuscript claims a corresponding result over certain fixed rings, rather than fields. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine sorting objects from an unlimited number of workshops into types, while imposing a size limit on one associated component. The claim resembles saying that this component-size limit leaves only finitely many types, even though the workshops themselves are unrestricted. The image does not capture the precise mathematical rule for deciding when two objects have the same type.
Why it matters
The claimed result gives the finiteness statement of Donovan's conjecture over each fixed algebraically closed field of positive characteristic. Its scope includes nonabelian defect groups and the prime . The companion extends the bounded-defect finiteness statement to a mixed-characteristic integral setting, including ramified rings.
2 · Going further
The field version fixes an algebraically closed field of characteristic and a bound on defect-group order. The question is whether blocks of finite groups satisfying that bound represent only finitely many -linear Morita equivalence classes. The main manuscript claims an affirmative answer for every such and every prime , with no abelian hypothesis on the defect groups.
The source identifies this statement as Donovan's conjecture. The supplied abstract states the result and its scope, but does not describe the argument, reductions, or constructions used to obtain it. Consequently, no main idea of the argument can be extracted from the supplied text without adding unsupported material.
The companion manuscript claims integral Donovan finiteness over the Witt-vector ring : for every prime and positive integer , blocks of all finite groups whose defect groups have order at most fall into finitely many Morita equivalence classes over that ring. It also claims the same bounded-defect finiteness over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic , including ramified rings. Thus the two manuscripts address parallel finiteness questions over fields and over mixed-characteristic rings; the abstracts do not specify a logical dependence between their arguments.
Useful notions
- Finite groups and their orders
- Fields and prime characteristic
- Blocks and defect groups
- Morita equivalence
- Complete discrete valuation rings and residue fields
What was known before / what is new
Before
The source presents the field finiteness question under the name Donovan's conjecture. It does not give earlier partial results, historical milestones, or previously established cases, so no more detailed account of prior knowledge is supported here.
New
The main manuscript claims bounded-defect Morita finiteness over every fixed algebraically closed field of positive characteristic. The companion claims the integral counterpart over Witt vectors and, more generally, over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field. Both statements explicitly allow nonabelian defect groups and include .
Limits and assumptions
The field is fixed and algebraically closed, its characteristic is a positive prime, and defect-group order is bounded. In the integral statement, the ring is fixed and is a complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p; ramification is allowed. The claims concern finiteness of Morita equivalence classes, not a stated enumeration or numerical bound on their number. They do not assert finiteness without a defect-order bound or a single finiteness statement across varying base fields or rings. The abstracts supply no argument and identify no remaining open cases outside these hypotheses. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Donovan's conjecture —
- In the field formulation given here, the assertion that bounding defect-group order leaves only finitely many Morita equivalence classes of blocks over each fixed algebraically closed field of positive characteristic.
- Morita equivalence —
- The equivalence relation used in these statements to sort blocks into the types whose finiteness is asserted.
- Defect group —
- The group associated with a block whose order is bounded in these finiteness statements.
- Integral Donovan finiteness —
- The companion manuscript's bounded-defect Morita finiteness assertion over the specified rings rather than fields.
Keywords : Donovan's conjecture · Morita equivalence · Defect groups
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Donovan's Conjecture over Algebraically Closed Fields
24/09/2026We 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.
Integral Donovan Finiteness over Witt Vectors
25/09/2026We 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 . 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.