Finitistic dimension of Artinian rings with vanishing radical cube
For left Artinian rings \(\Lambda\) with the property that the cube of the Jacobson radical \(J\) is zero, we confirm the longstanding conjecture that the supremum, \(\text{fin dim}(\Lambda)\), of the projective dimensions of those finitely generated left \(\Lambda\)-modules which have finite projective dimension, is finite. In fact, we prove the conjecture under various relaxed finiteness prerequisites for \(J^ 3\). A major portion of our argument is algorithmic in nature, permitting the computation of tight upper bounds for \(\text{fin dim}(\Lambda)\), based on information about the composition factors of the indecomposable projective left ideals of \(\Lambda\). The roughest form of the bounds obtained in case \(J^ 3=0\) (or more generally, for the case where all simple composition factors of the left \(\Lambda\)-module \(J^ 3\) have finite projective dimension) is as follows: \(\text{fin dim}(\Lambda)\leq 2m+d+1\), where \(m\) is the number of isomorphism types of simple left \(\Lambda\)-modules of infinite projective dimension and \(d\) is the supremum of the projective dimensions of those simple modules having finite projective dimension. An application of the results yields further classes of rings for which the Nakayama Conjecture is known to hold.
scientific article; zbMATH DE number 957
scientific article