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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-05-03801-8
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 \[ \textrm {H}=\mathbb {F}[h_1,\dotsc ,h_{n_0},h_{n_0+1}^p,\dotsc ,h_{n_1}^p, h_{n_1+1}^{p^2},\dotsc ,h_{n_e}^{p^e}],\] 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

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