A New Approximation Theory Which Unifies Spherical and Cohen-macaulay Approximations

In the late 1960s Auslander and Bridger [2] introduced a notion of approximation which they used to prove that every module whose n syzygy is n-torsionfree can be described as the quotient of an n-spherical module by a submodule of projective dimension less than n. About two decades later Auslander and Buchweitz [3] introduced the notion of Cohen-Macaulay approximation which they used to show that the category of finitely generated modules over a Cohen-Macaulay local ring with the canonical module is obtained by gluing together the subcategory of maximal Cohen-Macaulay modules and the subcategory of modules of finite injective dimension. Our purpose is to give a new approximation theorem, which unifies these two notions of [2] and [3]. Before stating our own result, let us briefly summarize the theorems of [2] and [3]. Throughout let R be a commutative Noetherian ring. Let M be a finitely generated R-module and n ≥ 1 an integer. Then we say that M is n-spherical if ExtR(M,R) = 0 for 1 ≤ i ≤ n. We say that M is n-torsionfree if the transpose TrM ofM is n-spherical. Let ΩM denote the n syzygy ofM . With this notation the approximation theorem of Auslander and Bridger is stated as follows.