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

Neusel, Mara

doi:10.1090/S0002-9947-05-03801-8

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups
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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 $n_i=\dim _{\mathbb {F}}(W_0\oplus \dotsb \oplus W_i)$.

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