Let (R,m) be a commutative Noetherian local ring with m = (0). We give a condition for R to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero with parameters in an open subset of projective space. We shall finally show that the subcategory consisting of modules of G-dimension zero over R is not necessarily a...

In this paper we consider the notion of the Weyl filtration dimension and good filtration dimension of modules for a linear algebraic group. These concepts were first introduced by Friedlander and Parshall [15] and may be considered a variation of the notion of projective dimension and injective dimension respectively. (The precise definition is given in 2.2.) The Weyl filtration dimension of a...

In the first part of this paper the projective dimension of the structural modules in the BGG category O is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig’s a-function is formulated (and proved for type A). The second part deals with the extension algebra of Verma modules. I...

‎we introduce a generalization of the notion of‎ depth of an ideal on a module by applying the concept of‎ local cohomology modules with respect to a pair‎ ‎of ideals‎. ‎we also introduce the concept of $(i,j)$-cohen--macaulay modules as a generalization of concept of cohen--macaulay modules‎. ‎these kind of modules are different from cohen--macaulay modules‎, as an example shows‎. ‎also an art...