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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Depth and cohomological connectivity in modular invariant theory


Authors: Peter Fleischmann, Gregor Kemper and R. James Shank
Journal: Trans. Amer. Math. Soc. 357 (2005), 3605-3621
MSC (2000): Primary 13A50, 20J06, 13C15
Published electronically: November 4, 2004
MathSciNet review: 2146641
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Abstract: Let $G$ be a finite group acting linearly on a finite-dimensional vector space $V$ over a field $K$ of characteristic $p$. Assume that $p$ divides the order of $G$ so that $V$ is a modular representation and let $P$ be a Sylow $p$-subgroup for $G$. Define the cohomological connectivity of the symmetric algebra $S(V^*)$ to be the smallest positive integer $m$ such that $H^m(G,S(V^*))\not=0$. We show that $\min\left\{\dim_K(V^P) + m + 1,\dim_K(V)\right\}$is a lower bound for the depth of $S(V^*)^G$. We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if $G$ is $p$-nilpotent and $P$ is cyclic, then, for any modular representation, the depth of $S(V^*)^G$is $\min\left\{\dim_K(V^P) + 2,\dim_K(V)\right\}$.


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Additional Information

Peter Fleischmann
Affiliation: Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, United Kingdom
Email: P.Fleischmann@kent.ac.uk

Gregor Kemper
Affiliation: Zentrum Mathematik - M11, Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany
Email: kemper@ma.tum.de

R. James Shank
Affiliation: Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, United Kingdom
Email: R.J.Shank@kent.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03591-3
PII: S 0002-9947(04)03591-3
Received by editor(s): July 17, 2003
Received by editor(s) in revised form: December 17, 2003
Published electronically: November 4, 2004
Additional Notes: This research was supported by EPSRC grant GR/R32055/01
Article copyright: © Copyright 2004 American Mathematical Society