Depth and cohomological connectivity in modular invariant theory
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- by Peter Fleischmann, Gregor Kemper and R. James Shank PDF
- Trans. Amer. Math. Soc. 357 (2005), 3605-3621 Request permission
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 $\min \left \{\dim _K(V^P) + 2,\dim _K(V)\right \}$.References
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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, 85 748 Garching, Germany
- MR Author ID: 608681
- Email: kemper@ma.tum.de
- R. James Shank
- Affiliation: Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, United Kingdom
- MR Author ID: 289797
- ORCID: 0000-0002-3317-4088
- Email: R.J.Shank@kent.ac.uk
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 3605-3621
- MSC (2000): Primary 13A50, 20J06, 13C15
- DOI: https://doi.org/10.1090/S0002-9947-04-03591-3
- MathSciNet review: 2146641