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
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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 $n_i=\dim _{\mathbb {F}}(W_0\oplus \dotsb \oplus W_i)$.References
- Nathan Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. MR 0172871, DOI 10.1007/978-1-4612-9872-4
- Mara D. Neusel, Invariants of some abelian $p$-groups in characteristic $p$, Proc. Amer. Math. Soc. 125 (1997), no. 7, 1921–1931. MR 1377000, DOI 10.1090/S0002-9939-97-03814-8
- Mara D. Neusel, Inverse invariant theory and Steenrod operations, Mem. Amer. Math. Soc. 146 (2000), no. 692, x+158. MR 1693799, DOI 10.1090/memo/0692
- Mara D. Neusel, Localizations over the Steenrod algebra. The lost chapter, Math. Z. 235 (2000), no. 2, 353–378. MR 1795513, DOI 10.1007/s002090000155
- Mara D. Neusel, Unstable Cohen-Macaulay algebras, Math. Res. Lett. 8 (2001), no. 3, 347–359. MR 1839483, DOI 10.4310/MRL.2001.v8.n3.a10
- Mara D. Neusel and Larry Smith, Invariant theory of finite groups, Mathematical Surveys and Monographs, vol. 94, American Mathematical Society, Providence, RI, 2002. MR 1869812, DOI 10.1086/342122
- Clarence Wilkerson, Classifying spaces, Steenrod operations and algebraic closure, Topology 16 (1977), no. 3, 227–237. MR 442932, DOI 10.1016/0040-9383(77)90003-9
- Clarence W. Wilkerson Jr., Rings of invariants and inseparable forms of algebras over the Steenrod algebra, Recent progress in homotopy theory (Baltimore, MD, 2000) Contemp. Math., vol. 293, Amer. Math. Soc., Providence, RI, 2002, pp. 381–396. MR 1890745, DOI 10.1090/conm/293/04957
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