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On the Cohen-Macaulay property of multiplicative invariants

Author: Martin Lorenz
Journal: Trans. Amer. Math. Soc. 358 (2006), 1605-1617
MSC (2000): Primary 13A50, 16W22, 13C14, 13H10
Published electronically: June 21, 2005
MathSciNet review: 2186988
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Abstract: We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group $\mathcal{G}$. By definition, these are $\mathcal{G}$-actions on Laurent polynomial algebras $\Bbbk[x_1^{\pm 1},\dots,x_n^{\pm 1}]$that stabilize the multiplicative group consisting of all monomials in the variables $x_i$. For the most part, we concentrate on the case where the base ring $\Bbbk$ is $\mathbb{Z} $. Our main result states that if $\mathcal{G}$ acts non-trivially and the invariant ring $\mathbb{Z} [x_1^{\pm 1},\dots,x_n^{\pm 1}]^\mathcal{G}$ is Cohen-Macaulay, then the abelianized isotropy groups ${\mathcal{G}}_m^{{ab}}$ of all monomials $m$ are generated by the bireflections in $\mathcal{G}_m$ and at least one ${\mathcal{G}}_m^{{ab}}$ is non-trivial. As an application, we prove the multiplicative version of Kemper's $3$-copies conjecture.

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Additional Information

Martin Lorenz
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Keywords: Finite group action, ring of invariants, multiplicative invariant theory, height, depth, Cohen-Macaulay ring, group cohomology, generalized reflections, bireflections, integral representation, binary icosahedral group
Received by editor(s): December 15, 2003
Received by editor(s) in revised form: May 26, 2004
Published electronically: June 21, 2005
Additional Notes: This research was supported in part by NSF grant DMS-9988756
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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