The finitistic dimension conjecture for finite-dimensional algebras

Theorem 1.1: there is a finite-dimensional complex algebra $A$ with, for each $m\ge1$, a finite-dimensional left module $N_m$ with $2m-2\le\mathrm{pd}\,N_m<\infty$; so $\mathrm{findim}\,A=\infty$ and the big finitistic dimension is infinite too, refuting the conjecture for Artin algebras. By Rickard's theorem injective left $A$-modules do not generate the derived category. Corollary (left-right asymmetry): an algebra $\Lambda$ with both finitistic dimensions infinite on the left and zero on the right. The construction passes through a finitely presented group algebra with Abels-type central involutions. A companion OpenAI manuscript (Tachikawa family) independently gives characteristic-two examples.

Theorem 1.1: there is a finite-dimensional complex algebra $A$ with, for each $m\ge1$, a finite-dimensional left module $N_m$ with $2m-2\le\mathrm{pd}\,N_m<\infty$; so $\mathrm{findim}\,A=\infty$ and the big finitistic dimension is infinite too, refuting the conjecture for Artin algebras. By Rickard's theorem injective left $A$-modules do not generate the derived category. Corollary (left-right asymmetry): an algebra $\Lambda$ with both finitistic dimensions infinite on the left and zero on the right. The construction passes through a finitely presented group algebra with Abels-type central involutions. A companion OpenAI manuscript (Tachikawa family) independently gives characteristic-two examples.