Cohen-Macaulay dimension for complexes
Let \(R\) be a commutative noetherian local ring. The projective dimension of an \(R\)-module is a well-known and widely studied numerical invariant in classical homological algebra whose finiteness can characterize regularity of \(R\). However, there exist several refinements and extensions of this dimension that provide more subtle information. Namely, complete intersection dimension, G-dimension, and Cohen-Macaulay dimension that can be used to characterize complete intersection rings, Gorenstein rings, and Cohen-Macaulay rings, respectively. These homological dimensions satisfy the following inequalities \N\[\N\textrm{CM-dim}_{R}(M) \leq \textrm{G-dim}_{R}(M) \leq \textrm{CI-dim}_{R}(M) \leq \textrm{pd}_{R}(M)\N\]\Nwith equality to the left of any finite quantity. Each of these homological dimensions has been extended to complexes of \(R\)-modules. The focus of this paper is on the exploration of Cohen-Macaulay dimension within the category of homologically finite \(R\)-complexes. Since the Cohen-Macaulay dimension is defined via quasi-deformations, it is interesting to be able to study this dimension through resolutions. Accordingly, the author presents a significant theorem that allows the computation of the Cohen-Macaulay dimension for a homologically finite complex of \(R\)-modules using its syzygies. As a crucial application of this theorem, she demonstrates that any homologically finite \(R\)-complex \(X\) of finite Cohen-Macaulay dimension possesses a finite Cohen-Macaulay resolution, i.e. a bounded \(R\)-complex \(G\) of finitely generated \(R\)-modules that is isomorphic to \(X\) in the derived category of \(R\) and consists of modules of Cohen-Macaulay dimension zero.
scientific article; zbMATH DE number 7858238