On homological dimensions

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 $G_K$-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.