On homological dimensions

Abstract

For finite modules over a local ring the general problem is considered of finding an extension of the class of modules of finite projective dimension preserving various properties. In the first section the concept of a suitable complex is introduced, which is a generalization of both a dualizing complex and a suitable module. Several properties of the dimension of modules with respect to such complexes are established. In particular, a generalization of Golod's theorem on the behaviour of GK-dimension with respect to a suitable module K under factorization by ideals of a special kind is obtained and a new form of the Avramov-Foxby conjecture on the transitivity of G-dimension is suggested. In the second section a class of modules containing modules of finite CI-dimension is considered, which has some additional properties. A dimension constructed in the third section characterizes the Cohen-Macaulay rings in precisely the same way as the class of modules of finite projective dimension characterizes regular rings and the class of modules of finite CI-dimension characterizes complete intersections.