ABSTRACT
This chapter discusses a commutative noetherian local ring with maximal ideal and residue field. The local ring has sometimes a canonical module, which is of great significance, even for non Cohen-Macaulay rings. The ring is known to play an important role in the homological conjectures, especially in the improved new intersection or canonical element conjecture. The ring also has an important role in the problem of the intersection dimension formula. The chapter proves the intersection dimension formula when the finite projective dimension module is an A-module of finite injective dimension and is the canonical module of a quotient ring. It also discusses some linkage of the projective dimension module in a Gorenstein local ring, and illustrates a necessary condition for the existence of the canonical module of a noetherian local domain.
