On suitable modules and G-perfect ideals

Alex Gerko

Abstract:

A module K over a commutative Noetherian local ring R is said to be suitable (see [1]) if HomR(K,K) ∼= R and ExtiR(K,K) = 0 for i > 0. Trivial examples are a free module of rank 1 and a canonical module. We recall that the type of a module M (denoted typeM) is by definition the dimension, over the residue field k ∼= R/m of R, of the first non-zero ExtiR(k,M). The type of a suitable module satisfies the condition
typeK ∗dimk(K/mK) = typeR. (1)
In [2] (where such modules are called semidualizing) an example of a ring of type 2i is constructed with suitable modules of all types permissible by condition (1) for every natural number i.