## 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

## 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 $n_i=\dim _{\mathbb {F}}(W_0\oplus \dotsb \oplus W_i)$.## References

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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
- Received by editor(s): September 18, 2003
- Received by editor(s) in revised form: June 22, 2004
- Published electronically: November 1, 2005
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2231868

Dedicated: Dedicated to Clarence W. Wilkerson on the occasion of his $60$th birthday