Systems of parameters and the Cohen-Macaulay property

Cohen-Macaulay rings play a central role in commutative algebra and there are many connections between systems of parameters and the Cohen-Macaulay property. In a Cohen-Macaulay ring, every system of parameters is also a regular sequence (roughly speaking it behaves like a set of polynomial variables). A classical result due to Rees says that when working in a Cohen-Macaulay ring, a certain class of modules of homomorphisms defined by systems of parameters is always isomorphic to a certain free module of rank one. Recently, K. Bahmanpour and R. Naghipour showed that in a non-Cohen-Macaulay ring, the same class of modules of homomorphisms is sometimes decomposable as a direct sum, and therefore is not a free module of rank one. In this talk, we will present stronger theorems in the non-Cohen-Macaulay case, and present illustrative examples about the decompositions obtained.