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Projective resolutions of representations of GL (n).

The representations of the algebraic group GL(n) are well understood over a field of characteristic 0, but they are much more complicated in characteristic p (or over the integers), essentially because there can be nontrivial extensions between two representations. Akin-Buchsbaum [2] and Donkin [10] improved matters by showing that the category P (n, r) of polynomial GL(F)-modules (n = dim V) which are homogeneous of fixed degree r, such äs ® , AV9 S"T, the irreducible subquotients of these modules, and so on, has finite global dimension. That is, there is an integer N depending on r and n = dim V such that every polynomial GL(F)-module of degree r has a resolution of length ^N by projective objects in the category of degree-r representations of CL(F). An equivalent Statement is that for polynomial representations A, B of degree r,