A pr 2 00 9 MODULES OF COVARIANTS IN MODULAR INVARIANT THEORY

Let the finite group G act linearly on the vector space V over the field k of arbitrary characteristic, and let H < G be a subgroup. The extension of invariant rings k[V ] ⊂ k[V ] is studied using modules of covariants. An example of our results is the following. Let W be the subgroup of G generated by the reflections in G. A classical theorem due to Serre says that if k[V ] is a free k[V ]-module then G = W . We generalize this result as follows. If k[V ] is a free k[V ]-module, then G is generated by H and W . Furthermore, the invariant ring k[V ] is free over k[V ] and is generated as an algebra by H-invariants and W -invariants.