REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: There are many conjectures from the representation theory of finite-dimensional algebras that have been transplanted to commutative algebra, and this process has enriched both fields significantly. An example is the celebrated Auslander-Reiten Conjecture, which states that a finitely generated module M over a finite-dimensional algebra A satisfying Extⁱ(M, M ⨁ A) = 0 for all i > 0 must be projective. This long-standing open conjecture is closely related to other important conjectures such as the Finitistic Dimension Conjecture from representation theory.

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, and interestingly enough, the counterexamples were obtained only after the AC conjecture had undergone the transplantation process to commutative algebra. Nonetheless, several classes of rings do satisfy Auslander's condition AC and remarkable homological properties of such classes have been uncovered recently; for example Christensen and Holm proved that the Auslander-Reiten Conjecture holds over Noetherian rings satisfying AC.

In this talk I will discuss a new condition, weaker than Auslander's condition AC, which also implies the Auslander-Reiten Conjecture for most commutative Noetherian local rings (one just needs to assume that the maximal ideal contains a non-zerodivisor.)  The talk is based on joint work with Ryo Takahashi [J. of Algebra, 382, 100-114, 2013]