Runar Ile: Cohen-Macaulay approximation in cofibred categories

Let A be a noetherian (and commutative) ring and N a (finite) A-module. If A is regular, then N has a finite projective resolution. Unfortunately this does not hold if A is singular. However, if A is Cohen-Macaulay the (sufficiently iterated) syzygy modules of N are maximal Cohen-Macaulay (MCM) modules. Hence N has a finite resolution with MCM modules. Auslander and Buchweitz’s theory for Cohen-Macaulay approximation of modules over Cohen-Macaulay rings with a canonical module gives a somewhat different solution to this problem. In particular they show that there is a short exact sequence 0 → L → M → N → 0 where M is MCM and L has finite injective dimension. After recapitulating their result, the talk will give an answer to the question:

If parameters are introduced in the equations defining N (and A), is there a natural way to introduce these parameters in M and L? Then conditions on N are given, which ensure stronger relations between the sets of parameterizations.