Finite resolutions of modules for reductive algebraic groups (Q1085278)

Let G be a reductive, connected, linear algebraic group over an algebraically closed field K. Let M be a rational G-module of finite dimension over K. Akin and Buchsbaum have constructed finite resolutions of some modules M by modules that are direct sums of tensor products of exterior powers of the natural representation. The main result here provides a characterization of those modules M which admit such a resolution. The validity of this result for Chevalley group schemes and general linear group schemes over a principal ideal domain is also discussed. The author applies his results to prove, amongst other things, that some modules appearing in the theory of representations of the general linear group (introduced by James) have an Akin-Buchsbaum resolution.