Title:

Modules over group algebras which are free on restriction to a maximal subgroup

Consider the following situation: k will be an algebraically closed field of characteristic p and G will be a finite pgroup, V will be a nonprojective, indecomposable kGmodule which is free on restriction to some maximal subgroup of G. Our purpose in doing this is to investigate Chouinard's theorem  all the proofs of which have been cohomological in nature  in a representationtheoretic way. This theorem may be shown to be equivalent to saying that, if G is not elementary abelian, V cannot be free on restriction to all the maximal subgroups of G. It is shown how to construct an exact sequence: O → V → P → P → V → O with P projective. From this an almost split sequence, O → V → X → V → O is constructed. It is shown that X can have at most two indecomposable summands. If φ denotes the Frattini subgroup of G, then V is free on restriction to φ. We may regard the set of φfixed points of V, V̄, as a module for Ḡ =G/φ. But Ḡ is elementary abelian, so we may consider the Carlson variety, Y(V̄)  this may be regarded as a subset of J/J² where J denotes the augmentation ideal of kG. It is shown that Y(V̄) is always a line. We define YG to be the union of all the lines Y(V̄) as V runs over all the kGmodules with the properties above. It is shown that YG is the whole of J/J² if and only if G is elementary abelian. It is also shown that, when G is one of a particular class of pgroups  the pseudospecial groups  which form the minimal counterexamples to Chouinard's theorem, that YG is the set of zeros of a sequence of homogeneous polynomials with coefficients in the field of p elements. Indeed, a specific construction for these polynomials is given.
