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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups
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by Mara D. Neusel PDF
Trans. Amer. Math. Soc. 358 (2006), 4689-4720 Request permission


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:
  • 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
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 4689-4720
  • MSC (2000): Primary 55S10, 13A50, 13-xx, 55-xx
  • DOI:
  • MathSciNet review: 2231868