Relative trace ideals and Cohen-Macaulay quotients of modular 
invariant rings

Peter Fleischmann
Institute for Experimental Mathematics
University of Essen
Ellernstr. 29, 45326 Essen, Germany

The paper is to appear in 
`Proceedings of the Euroconference: Computational Methods for
Representations of Groups and Algebras', ed. P. Draexler, G.O. Michler,
C.M. Ringel, Birkhaeuser (Basel)

In this paper I investigate `relative trace ideals' A_X^G of p - 
modular invariant rings A, which are defined in analogy to modular 
representation theory, as images of relative trace maps from invariants
of subgroups in a family X.
It is shown that if X is a family of proper subgroups of a fixed
Sylow p - group P of G, then the A_X^G is always a proper 
ideal of A^G with height bounded above by the codimension of 
the fixed point space V^P.
We also prove that if V is relatively X - projective, then A_X^G
still contains all invariants of degree not divisible by p. 
I also give a `geometric analysis' of trace ideals, in particular of the 
`largest' trace ideal A_<P^G:= \sum_{Q<P}A_Q^P. I prove that 
I^{G,P} := \sqrt{A^G_<P} is a prime ideal which has the geometric 
interpretation as `vanishing ideal' of G - orbits with length coprime to p.
It is shown that the quotient A^G/I^{G,P} is always a Cohen - Macaulay algebra,
if the action of P is defined over the prime field. This generalizes the well 
known result of Hochster and Eagon (for the case P=1) that A^G is
Cohen - Macaulay in the coprime case.