{"id":"openai-math/A-counterexample-to-Tachikawas-second-conjecture","pdf":"https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026","tags":["representation-theory","symmetric-algebras","homological-algebra","nakayama-conjecture"],"score":0,"title":"A counterexample to Tachikawa's second conjecture","domain":"Algebra","family":"199","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.RA","summary":"We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over $`k=\\mathbb F_2(q,H_1,H_2)`$ with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k .","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":2,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://www.math.uni-bielefeld.de/~ringel/new.html","kind":"web","site":"math.uni-bielefeld.de","text":"A new age has started! (9.10.2026) It is said that AI has exhibited examples which show that all the classical homological conjectures are wrong. Whether these assertions are correct, has still to be verified (like any claim of a mathematician). But at least someone has started to take…","added":"2026-11-03","title":"C.M.Ringel (Bielefeld): New Activities","author":"Claus Michael Ringel"},{"url":"https://sites.google.com/view/rene-marczinzik/homological-conjectures-and-problems#:~:text=After%20OpenAI%20solved","kind":"web","site":"sites.google.com","text":"After OpenAI solved (it still has to be checked carefully!) the second Tachikawa, most of the important homological conjectures in our field might be wrong after all. Here I suggest some own interesting problems and conjectures related to homological algebra of finite dimensional algebras. (last…","added":"2026-11-03","title":"Rene Marczinzik - Homological conjectures and problems","author":"Rene Marczinzik"}],"formalized":true,"family_title":"Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures"}