Relative Trace Ideals and Cohen-Macaulay Quotients of Modular Invariant Rings

Abstract

Let G be a finite group, F a field whose characteristic p divides the order of G and A ^G the invariant ring of a finite-dimensional FG-module V. In nalogy to modular representation theory we define for any subgroup H ≤ G the (relative) trace-ideal A ^G _H ⊲ A ^G to be the image of the relative trace map $t_H^G:{A^H} \to {A^G},f \mapsto \sum\nolimits_{g \in [G:H]} {g(f)}$. Moreover, for any family χ of subgroups of G, we define the relative trace ideals $A_\chi ^G: = \sum\nolimits_{X \in \chi } {A_X^G \triangleleft {A^G}}$ and study their behaviour.

If χ consists of proper subgroups of a Sylow p-group P ≤ G, then A ^χ _G is always a proper ideal of A ^G; in fact, we show that its height is bounded above by the codimension of the fixed point space V ^P. But we also prove that if V is relatively χ-projective, then A ^χ _G still contains all invariants of degree not divisible by p. If V is projective then this result applies in particular to the (absolute) trace ideal A ^{e} _G .

We also give a ‘geometric analysis’ of trace ideals, in particular of the ideal $A_{ < P}^G: = \sum\nolimits_{Q < P} {A_{{Q^O}}^G}$, and show that ${\mathcal{I}^{G,P}}: = \sqrt {A_{ < P}^G}$ is a prime ideal which has the geometric interpretation as ‘vanishing ideal’ of G-orbits with length coprime to p. It is shown that ${A^G}/{\mathcal{I}^{G,P}}$ is always a Cohen-Macaulay algebra, if the action of P is defined over the prime field. This generalizes a well known result of Hochster and Eagon for the case P = 1 (see [13]). Moreover we prove that if V is a direct summand of a permutation module (i.e. a ‘trivial source module’), then the A ^χ _G are radical ideals and $A_{ < P}^G/{\mathcal{I}^{G,P}}$. Hence in this case the ideal and the corresponding Cohen-Macaulay quotient can be constructed using relative trace maps.