An algebra of infinite little finitistic dimension

OpenAI's counterexample (2026) to the finitistic dimension conjecture: a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. A finitely presented group algebra with a selection process of unbounded extinction is realized by a single complex of bimodules over a three-vertex algebra, simulated by an ordinary bimodule, and turned into projective dimension by a trivial extension.

Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Finitistic dimensions and minimal resolutions
2. A selection process with unbounded extinction
3. Directed algebras and quadratic presentations
4. A Verdier quotient and an odd double
5. Clearing denominators in a finite diagram
6. A bimodule complex that realizes selection
7. Simulation by an ordinary bimodule
8. Trivial extensions and unbounded projective dimension

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Words, free groups and Cayley graphs (Van Kampen diagrams)
• Derived categories, derived tensor products and Verdier quotients of modules over finite-dimensional algebras.
• Group presentations.