Auslander-type conditions and weakly Gorenstein algebras (Q6647811)

A ring \(R\) is said to satisfy the Auslander condition if and only if the flat dimension of the \(n\)-term in a minimal injective resolution of \(R\) is at most \(n-1\). The motivation behind this condition comes from one of the most influential works of Bass. Indeed, \textit{H. Bass} proved in [Math. Z. 82, 8--28 (1963; Zbl 0112.26604)] that a commutative ring is Iwanaga-Gorenstein if and only if it satisfies the Auslander condition. Later, Auslander proved that the above condition is left-right symmetric and together with Reiten, they conjectured that an Artin algebra is Iwanaga-Gorenstein if and only if it satisfies the Auslander condition.\N\NThis homological conjecture remains an open problem and several generalisations of the Auslander condition have been proposed. In particular, one can consider rings \(R\) for which the flat dimension of the \(n\)-term in a minimal injective resolution of \(R\) as left module (resp. as right module) is at most \(n-1+m\) for some \(m\geq 0\) (resp. \(n-1+m'\) for some \(m'\geq 0\)). Those rings are said to have an Auslander-type condition (which is not necessarily left-right symmetric).\N\NGiven a ring \(R\) satisfying this Auslander-type condition, the author shows that \(R\) is Iwanaga-Gorenstein if and only if \(R\) is weakly Gorenstein if and only if \(R\) is left (or right) Iwanaga-Gorenstein. In particular, the author obtains that a ring satisfying the Auslander condition is Iwanaga-Gorenstein if and only if it is weakly Gorenstein.