Bounds on Global and Finitistic Dimension for Finite Dimensional Algebras with Vanishing Radical Cube Original Research Article

It is shown that, given any left artinian ring Λ which has vanishing radical cube and n isomorphism classes of simple left modules, the global dimension of Λ is either infinite or bounded above by n2 − n, and the left finitistic dimension of Λ is always less than or equal to n2 + 1; in fact, sharper bounds are obtained, but these are not as easily described. Moreover, a tight grid for the "distribution" of the projective dimensions of the simple left Λ-modules, again in terms of n, is set up. The key to these estimates is a sequence of matrix groups which stores homological information on Λ in an efficient form. Additional applications of this machinery include lower bounds on the projective dimensions of the simple left Λ-modules and hence on the global dimension of Λ.