The relative trace ideal and the depth of modular rings of invariants
Let \(G\) be a finite group, \(p\) a prime divisor of the order of \(G\), \(k\) a field of characteristic \(p\), and \(V\) a finitely generated \(kG\)-module. Consider the algebra \(A=k[V]\) and the subalgebra of invariants \(A^G\). The depth of \(A^G\) is the length of a maximal regular sequence in \(A^G\). Since \(V\) is a modular representation of \(G\), the algebra \(A^G\) may fail to be Cohen-Macaulay, i.e., the depth might be less than the Krull dimension. The authors prove that the depth of \(A^G\) is the sum of the dimension of the fixed point space of the \(p\)-Sylow subgroup and the grade of the relative trace ideal. They also determine which of the Dickson invariants lie in the radical of the relative trace ideal and describe how to use the Dickson invariants to compute the grade of the relative trace ideal.
scientific article; zbMATH DE number 1966457