Let k be a field and G a finite group, acting on the polynomial ring A := k[x₁,…,x_d] by graded k-algebra automorphisms. The ring of invariants A^G := {f∈A  |  g(f) = f} is the main object of study in Invariant Theory. The theory is very well developed in the “classical case”, where the characteristic of k is zero, but far less so in the case of positive characteristic, in particular the “modular case”, where the characteristic divides the group order |G|.

In that case there are open questions about the constructive complexity of A^G, measured by degree bounds for generators, and about the structural complexity, measured by the depth (=length of maximal regular sequence, or “cohomological co-dimension”) of A^G as a module over a homogeneous system of parameters.

In my talk I will, after a brief introduction, report on some recent results dealing with both types of questions.