This paper builds upon the work of Cline [C] and Donkin [D1] to describe explicit equivalences between some categories associated to the category of rational modules for a reductive group G and categories associated to the category of rational modules for a Levi subgroup H . As an application, we establish an Ext-transfer result from rational G-modules to rational H-modules. In case G = GLn, th...

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...

Introduction Let G be a simply connected and connected semisimple group over an algebraically closed field k of characteristic p > 0. T ⊂ G is a maximal torus and R is the root system relative to T . X(T ) is the weight lattice. Let B ⊃ T be a Borel subgroup corresponding to the negative roots R− = R. Denote by Gr the r-th Frobenius kernel of G. The socle and radical structures of the cohomolog...

We introduce the notion of E-depth graded modules over polynomial rings to measure depth certain Ext modules. First, we characterize with (sufficiently) large as those whose partial general initial submodules preserve Hilbert function local cohomology supported at irrelevant maximal ideal, extending a result Herzog and Sbarra on sequentially Cohen-Macaulay Second, describe cone tables sufficien...

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...

We give an explicit algorithm to compute a projective resolution of a module over the noncommutative ring based on the noncommutative Gröbner bases theory. Introduction LetK be a field and Γ be a ring overK. We generally assume that Γ has a unit, ǫ : K → Γ, as well as an augmentation η : Γ → K. For graded Γ-module M and N , Ext Γ (M,N) is defined with a projective resolution of M . This Ext-fun...

Let A A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the internal ...

In this paper we study the Cartan matrix associated to Ext-projective stratifying system induced by a basic and τ-rigid object M in mod(A) means of g-vectors indecomposable direct summands M. particular show that group module can be calculated directly using these vectors. Moreover characterise systems coming from modules have diagonal matrix.