On the Good Filtration Dimension of Weyl Modules for a Linear Algebraic Group

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 module is always at most its projective dimension. In fact, it is often much less. In the situation of algebraic groups the Weyl and good filtration dimensions are always finite for a finite dimensional module (unlike the projective and injective dimensions which are usually infinite). Thus knowing these dimensions give us another tool for calculating the cohomology of an algebraic group. Indeed we use knowledge of these dimensions to calculate various Ext groups for G. We had previously calculated the good filtration dimension of the irreducible modules for S(n, r), the Schur algebra corresponding to GLn(k) when n = 2 and n = 3 in [21]. We were then able to determine the global dimension of S(n, r). The proof in [21] relies heavily on the use of filtrations of the induced modules ∇(λ), λ a dominant weight, by modules of the form ∇(μ) ⊗ L(ν). In this paper we instead use the translation functors introduced by Jantzen to calculate properties of the induced modules and the Weyl modules (denoted ∆(λ)) for an algebraic group. We first calculate the Weyl filtration dimension (abbreviated wfd), of the induced modules for regular weights (theorem 4.2). We then prove Ext (