Homological dimensions of the Jacobson radical (Q6572983)

The authors mainly investigate the homological properties of the Jacobson radical \(J\), and its higher syzygies, over a semiperfect noetherian ring \(A\). Particularly, they prove that if \(A\) is a semiperfect noetherian ring, then the injective dimension of \(J\) coincides with the global dimension of \(A\). As \(J\) is the first syzygy of \(A/J\), this inspires the authors to consider the higher syzygies of \(A/J\). For a minimal Auslander-Gorenstein algebra \(A\), they prove that the injective dimension of any non-zero higher syzygy of \(A/J\) and the global dimension of \(A\) are identical.