Homological dimension in local rings

Introduction. This paper is devoted primarily to the study of commutative noetherian local rings. The main task is to compare purely algebraic properties with properties of a homological nature. A large part of this paper is an elaboration of [2](2) which contained no proofs. We use [3] as a reference source for homological algebra. We begin with a list of the most important notions and an outline of results. Unless stated otherwise, we assume throughout this paper that all rings are commutative noetherian rings with identity elements and that all modules are finitely generated and unitary. If R is a local ring, we shall denote its maximal ideal by m and the quotient field R/m by F. Given a ring R and an l?-module E we denote by hdRE the integer (finite or + co) which in [3] is denoted by dim# E. We have hdR E^nil there exists an exact sequence