| Description |
First, we recall some classical results from invariant theory, and the direct summand property of ring extensions. We review the local cohomology functors and the F-signature of a ring. We consider the question of how many independent splittings the ring of invariants of a finite group action has; equivalently, what the F-signature of the invariant ring is. In particular, we consider the question of when the ring of invariants of a finite group G-action on a vector space over a field of positive characteristic p > 0, where p divides G, is a direct summand of the polynomial ring. We prove that if the a-invariant of the ring of invariants is equal to that of the polynomial ring, then it is not a direct summand. We provide further evidence for a conjecture of A. Broer related to this question. Following the work of Watanabe-Yoshida, A. Singh, and M. Von Korff, we study the F-signature of affine toric varieties. We determine which are toric varieties of a particular dimension have the largest F-signature, and analyze the structure of the set of values. Next, we study the separating rank of a finite group action - the least number of invariants required to separate the orbits of the group action. We find a lower bound on the separating rank in terms of the ranks of generators of stabilizer subgroups of the action. This result is a generalization of a theorem of Serre on when rings of invariants are polynomial rings. We show that the lower bound is sharp for large classes of examples. This part is based on joint work with Emilie Dufresne. We end by posing a question on the vanishing of local cohomology that implies a generalization of the Shephard-Todd theorem. |
