Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures

The explanation

The heart of the proof is a small two-by-two table of numbers. Its rows are H₁ to the power −m, then 1, and H₂ to the power −m, then 1. Its determinant is H₁^(−m) − H₂^(−m). The table can be undone exactly when that is not zero. As a stand-in, try H₁ = 2 and H₂ = 3. For m = 1, 2, 3 the determinant is 1/6, 5/36 and 19/216. It is never zero, because a power of 2 never equals the same power of 3.

Now try H₁ = −1 and H₂ = 1. For m = 2 the determinant is 1 − 1 = 0, and the table cannot be undone. The ratio −1 returns to 1 after two steps. The paper picks H₁ and H₂ whose ratio never returns to 1Source 1. Then the table can be undone for every m at onceSource 2. That is what removes every obstruction in the proof.

Some words first. An algebra here is a finite-dimensional space of numbers with a multiplication, such as square matrices. A module is something the algebra acts on, like vectors under matrices. Projective modules are the basic building blocks: pieces of free copies of the algebra. Ext groups measure the ways two modules can be glued together, one for each degree 1, 2, 3, and so on.

The Auslander–Reiten conjecture asks this. If a module has no gluings with itself or with the algebra, in any positive degree, must it be projectiveSource 3? It comes from Maurice Auslander and Idun Reiten's 1975 work on the generalized Nakayama conjectureSource 4. It was proved in many cases. These include large classes of commutative ringsSource 5. They also include algebras whose radical cubed is zeroSource 6. Schulz found a module with no self-gluings, but over a skew field, outside this settingSource 7.

The first paper claims a counterexampleSource 8. The field is built from arithmetic where 1 + 1 = 0, extended by three independent unknowns q, H₁ and H₂. Over it the paper builds a finite-dimensional algebra and a module Z. The module is not projective, yet all its positive-degree gluings with itself and with the algebra vanishSource 8. Z is also Gorenstein-projective, so a second conjecture about such modules fails tooSource 9. The examples survive every enlargement of the fieldSource 10.

How do the gluings vanish? The paper starts from a ten-dimensional algebra whose gluings form a very sparse pattern. It twists that algebra in two ways, scaled by H₁ and by H₂. Each degree-3m gluing gets multiplied by H₁^(−m) or by H₂^(−m). The two-by-two table compares the two twists. Because it can always be undone, every gluing of positive degree cancels, while the module stays non-projective.

The second paper aims at a stricter target. A self-injective algebra is one where the algebra itself has a strong symmetry. Tachikawa's second conjecture says a module with no positive self-gluings over such an algebra must be projectiveSource 11. The paper builds a symmetric algebra, a special self-injective kind, with such a module that is not projectiveSource 12. The extra difficulty is that gluings in negative degrees must also vanishSource 13.

From this second example the paper builds another algebra. That algebra refutes the classical Nakayama conjectureSource 14. It also refutes the Auslander–Gorenstein conjectureSource 15. It refutes the Wakamatsu tilting conjecture tooSource 16. The commutative version of these questions is not settled by either constructionSource 17.

How sure is this? A computer proof checker called Lean has checked formal statements of both main counterexamples. For Auslander–Reiten, it includes the eight simple modules and survival under field extension. For Tachikawa, it covers the symmetric algebra and its module. Lean does not include the later Nakayama-type consequences. No specialist review of these papers has been published.

What it doesn't show. This does not settle the commutative version of these conjectures.