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

 

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups


Author: Mara D. Neusel
Journal: Trans. Amer. Math. Soc. 358 (2006), 4689-4720
MSC (2000): Primary 55S10, 13A50, 13-xx, 55-xx
Published electronically: November 1, 2005
MathSciNet review: 2231868
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Abstract: We consider purely inseparable extensions $ \textrm{H}\hookrightarrow \sqrt[\mathscr{P}^*]{\textrm{H}}$ of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group $ G\le\textrm{GL}(V)$ and a vector space decomposition $ V=W_0\oplus W_1\oplus\dotsb\oplus W_e$ such that $ \overline{\textrm{H}}=(\mathbb{F}[W_0] \otimes\mathbb{F}[W_1]^p\otimes\dotsb\otimes\mathbb{F}[W_e]^{p^e})^G$ and $ \overline{\sqrt[\mathscr{P}^*]{\textrm{H}}}=\mathbb{F}[V]^G$, where $ \overline{(-)}$ denotes the integral closure. Moreover, $ \textrm{H}$ is Cohen-Macaulay if and only if $ \sqrt[\mathscr{P}^*]{\textrm{H}}$ is Cohen-Macaulay. Furthermore, $ \overline{\textrm{H}}$ is polynomial if and only if $ \sqrt[\mathscr{P}^*]{\textrm{H}}$ is polynomial, and $ \sqrt[\mathscr{P}^*]{\textrm{H}}=\mathbb{F}[h_1,\dotsc,h_n]$ if and only if

$\displaystyle \textrm{H}=\mathbb{F}[h_1,\dotsc,h_{n_0},h_{n_0+1}^p,\dotsc,h_{n_1}^p, h_{n_1+1}^{p^2},\dotsc,h_{n_e}^{p^e}],$

where $ n_e=n$ and $ n_i=\dim_{\mathbb{F}}(W_0\oplus\dotsb\oplus W_i)$.


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Additional Information

Mara D. Neusel
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042
Email: Mara.D.Neusel@ttu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03801-8
PII: S 0002-9947(05)03801-8
Keywords: Inseparable extensions, inseparable closure, Cohen-Macaulay, $\Delta$-relation, $\Delta_s$-relation, derivation, restricted Lie algebra, Steenrod algebra, Dickson algebra, invariant rings of finite groups.
Received by editor(s): September 18, 2003
Received by editor(s) in revised form: June 22, 2004
Published electronically: November 1, 2005
Dedicated: Dedicated to Clarence W. Wilkerson on the occasion of his $60$th birthday
Article copyright: © Copyright 2005 American Mathematical Society