{"id":"openai-math/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite","pdf":"https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026","tags":["group-algebras","ring-theory","finite-presentations"],"score":0,"title":"A Torsion-Free Group Algebra That Is Not Directly Finite","domain":"Algebra","family":"197","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.RA","summary":"We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-10-04","reactions":{},"discussion":[{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=Later%2C%20in","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"Later, in “A Torsion-Free Group Algebra That Is Not Directly Finite” (dated October 4), OpenAI claims to have found a counterexample to Kaplansky’s direct finiteness conjecture for group algebras over the field with two elements. It follows trivially that the latter document provides a…","added":"2026-11-04","title":"100+ reactions to 100+ solutions","author":"Johan Öinert"}],"formalized":false,"family_title":"A torsion-free group algebra that is not directly finite"}