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

Fleischmann, P. (1997) Relative trace ideals and Cohen-Macaulay quotients of modular invariant rings. In: Euroconference on Computational Methods for Representations of Groups and Algebras, APR 01-05, 1997 , Essen, Germany. (The full text of this publication is not available from this repository)

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Let G be a finite group, IF a field whose characteristic p divides the order of G and AG the invariant ring of a finite-dimensional FG-module V. In analogy to modular representation theory we define for any subgroup H less than or equal to G the (relative) trace-ideal A(H)(G) del a A(G) to be the image of the relative trace map t(G)(H) : A(H) --> A(G), f bar right arrow Sigma(g is an element of[G:H]) g(f). Moreover, for any family chi of subgroups of G, we define the relative trace ideals A behaviour. If chi consists of proper subgroups of a Sylow p-group P less than or equal to G, then A(chi)(G) is always a proper ideal of AG; in fact, we show that its height is bounded above by the codimension of the fixed point space VP. But We also prove that if V is relatively X-projective, then A(chi)(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) := Sigma(Q<P) A(Q)(G)0, and show that I-G,I-P := root(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)/I-G,I-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(chi)(G) are radical ideals and A(<P)(G) = I-G,I-P. Hence in this case the ideal and the corresponding Cohen-Macaulay quotient can be constructed using relative trace maps.

Item Type: Conference or workshop item (Paper)
Additional information: This item is a Proceedings Paper.
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Statistics
Depositing User: M. Nasiriavanaki
Date Deposited: 02 Jul 2009 07:18
Last Modified: 16 Apr 2014 13:11
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