1 · The essentials
The manuscript gives a counterexample to a proposed limit on mathematical complexity. It constructs a system of addition and multiplication over the complex numbers that has only finitely many independent directions. It then studies finite-dimensional spaces on which this system acts. These spaces have a measure of complexity based on how long a particular kind of mathematical construction takes. For the constructed system, this measure can be finite yet exceed any fixed bound. This contradicts the proposed limit, and the main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine one fixed workshop making objects that each require finitely many assembly stages, but with no upper limit on how many stages an object might need. The distinction is between every selected job eventually finishing and there being one deadline that covers them all.
Why it matters
The result distinguishes having finite complexity from having a uniform bound on finite complexity. According to the supplied abstract, the constructed algebra disproves the little finitistic-dimension conjecture. The counterexample already occurs for a finite-dimensional algebra over the complex numbers.
2 · Going further
The precise question concerns the projective dimensions of finite-dimensional modules over a finite-dimensional algebra. The little finitistic dimension is the supremum of those projective dimensions that are finite. The manuscript constructs a finite-dimensional complex algebra for which this supremum is infinite: its finite-dimensional modules have unbounded finite projective dimensions. Thus the issue is not merely the existence of a module with infinite projective dimension, but the absence of a bound among the finite values themselves.
The abstract presents the result as a counterexample to the little finitistic-dimension conjecture. Its stated approach is constructive: exhibit an algebra with the required unboundedness. However, the supplied text gives neither the defining relations of the algebra nor the modules or argument establishing their projective dimensions, so the mechanism cannot be explained further from this source. The main manuscript is marked as Lean-formalized; no companion manuscripts or more detailed description of the formalization's scope are supplied.
Useful notions
- Complex numbers and vector spaces
- Finite-dimensional algebras
- Modules over an algebra
- Projective resolutions and projective dimension
- Suprema and unbounded sets
What was known before / what is new
Before
The source identifies the little finitistic-dimension conjecture as the finiteness claim being challenged. It supplies no historical account, earlier partial results, or attribution for the conjecture.
New
The manuscript constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. The abstract states that this disproves the little finitistic-dimension conjecture. The main result has a Lean formalization.
Limits and assumptions
The stated construction concerns a finite-dimensional algebra over the complex numbers and finite-dimensional modules with finite projective dimension. Unboundedness among these finite values is the essential conclusion; the source does not say that every module has finite projective dimension. Although the main result is Lean-formalized, the supplied text does not specify the formalization's detailed scope. It also gives no explicit construction, proof details, companion results, or remaining open questions.
Mini glossary
- Finite-dimensional complex algebra —
- An algebra over the complex numbers whose underlying vector space has a finite basis.
- Module —
- A space equipped with an action of an algebra compatible with addition and multiplication.
- Projective resolution —
- An exact sequence of projective modules used to describe a module.
- Projective dimension —
- The shortest possible length of a projective resolution of a module, or infinity if no finite such resolution exists.
- Little finitistic dimension —
- The supremum of the finite projective dimensions attained by finite-dimensional modules over the algebra.
- Unbounded finite projective dimensions —
- Finite projective dimensions that exceed any proposed finite upper bound.
Keywords : Finite-dimensional algebra · Projective dimension · Little finitistic dimension
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)
Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
An algebra of infinite little finitistic dimension
23/09/2026We construct a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.