Finitistic Dimension Conjecture: OpenAI Claims a Counterexample

About this video

Every module over a finite-dimensional algebra can be resolved by projective modules, and the number of steps is its projective dimension, which may be finite or infinite. The little finitistic dimension conjecture, going back to Bass in 1960, says that for each such algebra the finite projective dimensions of its finitely generated modules are bounded: there is always a ceiling. OpenAI claims a counterexample: a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This is OpenAI's preprint, not yet peer reviewed. The animation computes the real projective resolutions of all 34 indecomposable modules of a Nakayama algebra on a loop of 7 arrows (16 never stop, the finite ones top out at 8); the tall towers at the end are a schematic of "no ceiling", not OpenAI's algebra.

Paper (OpenAI, catalogue result 198):
- An algebra of infinite little finitistic dimension: github.com/openai/math/blob/main/preprints/An-algebra-of-infi…

The animation illustrates the result; it is not a proof.

Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/198