On a conjecture of Auslander and Reiten, I

There are many conjectures from the representation theory of finite-dimensional algebras that have been transplanted into commutative algebra, and this process has significantly enriched both fields. A notable example is the Auslander–Reiten Conjecture, which asserts that a finitely generated module M over a finite-dimensional algebra A is projective if the Ext groups $Ext^n(M,M)$ and $Ext^n(M,A)$ vanish for all positive integers n. This long-standing conjecture is closely connected to other important conjectures in representation theory, including the Finitistic Dimension Conjecture and the Tachikawa Conjecture.

Although the Auslander–Reiten Conjecture originates in the representation theory of algebras, it has attracted considerable interest within commutative algebra. In 1994, Huneke and Wiegand formulated a conjecture about tensor products of torsion-free modules over one-dimensional commutative Noetherian integral domains. This conjecture, which remains unresolved, implies the Auslander–Reiten Conjecture for a broad class of commutative rings.

Auslander defined a condition, denoted by (AC), to analyze the Finitistic Dimension Conjecture, and conjectured that every finite-dimensional algebra satisfies (AC). This conjecture turned out to be false; interestingly, the first counterexamples were obtained only after the (AC) conjecture had been transplanted into commutative algebra. Nonetheless, several classes of rings do satisfy Auslander's condition (AC), and remarkable homological properties of such classes have been uncovered. For example, Christensen and Holm proved that the Auslander–Reiten Conjecture holds over Noetherian rings satisfying (AC).

In this talk, I will survey some of the literature on these problems and explore the relationship between the Huneke–Wiegand Conjecture, the Auslander–Reiten Conjecture, and related problems on the vanishing of Tor.