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

Fleischmann, Peter (1997) Relative trace ideals and Cohen-Macaulay quotients of modular invariant rings. In: Draxler, P. and Michler, G.O. and Ringel, C.M., eds. Computational Methods for Representations of Groups and Algebras. Progress In Mathematics, 173 . Birkhauser, Basel, Switzerland, pp. 211-233. ISBN 978-3-0348-9740-2. E-ISBN 978-3-0348-8716-8. (doi:10.1007/978-3-0348-8716-8_12) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

 The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. (Contact us about this Publication) Official URLhttp://dx.doi.org/10.1007/978-3-0348-8716-8_12

## 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 $$H\leq G$$ the (relative) trace-ideal $$A_H$$ $$^G\triangleleft A^G$$ to be the image of the relative trace map $$t^G_H : A^H \to A^G, f\mapsto \sum_{g\in [G:H]} g(f)$$. Moreover, for any family $$\chi$$ of subgroups of $$G$$, we define the relative trace ideals $$A^G_{\chi} := \sum_{X\epsilon\chi} A^G_X\triangleleft A^G$$ and study their behaviour.

If $$\chi$$ consists of proper subgroups of a Sylow $$p$$-group $$P\leq G$$ and then $$A$$ $$_G$$ $$^{\chi}$$ 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 (\chi\)-projective, then $$A$$ $$_G$$ $$^{\chi}$$ 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 $$A$$ $$_G$$ $$^{\lbrace e\rbrace}$$.

We also give a 'geometric analysis' of trace ideals, in particular of the ideal $$A^G_{ < P} :=\sum_{Q < P} A^G_{Q^O}$$ and show 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 $$A^G/I^{G,P}$$ 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 $$A$$ $$_G$$ $$^{\chi}$$ are radical ideals and $$A^G_{< p}/I^{G,P}$$. Hence in this case the ideal and the corresponding Cohen-Macaulay quotient can be constructed using relative trace maps.

Item Type: Book section 10.1007/978-3-0348-8716-8_12 This item is a Proceedings Paper. Finite Group; Prime Ideal; Direct Summand; Trace Ideal; Permutation Module Q Science > QA Mathematics (inc Computing science) Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Statistics M. Nasiriavanaki 02 Jul 2009 07:18 UTC 09 Sep 2019 08:51 UTC https://kar.kent.ac.uk/id/eprint/17326 (The current URI for this page, for reference purposes)