For $\mathcal{G}_{}$ some groupoid whose objects are the sub-vector spaces of a $\mathbb{F}_p$-vector space $W$, we define $H^*(W)^{\mathcal{G}_{}}$ a $nil$-closed, noetherian, unstable sub-algebra of $H^*(W)$ over the Steenrod algebra. The application on the appropriate ordered set of groupoids, that maps $\mathcal{G}_{}$ to $H^*(W)^{\mathcal{G}_{}}$ defines an isomorphism of posets to the set of noetherian, $nil$-closed, unstable sub-algebras of $H^*(W)$ of transcendence degree $\dim (W)$, ordered by inclusion.
Since any noetherian and integral unstable algebra of transcendence degree $\dim (W)$ admits an injection into $H^*(W)$, any such $nil$-closed unstable algebra is isomorphic to some $H^*(W)^{\mathcal{G}_{}}$.
We prove that $\mathcal{G}_{}$ encodes the centre, in the sense of Heard, of $H^*(W)^{\mathcal{G}_{}}$. Also, there is a $H^*(C)$-comodule structure on $K$ that is associated with the centre of $K$. For $K=H^*(W)^{\mathcal{G}_{}}$, we explain how the sub-algebra of primitive elements of $H^*(W)^{\mathcal{G}_{}}$ for this comodule structure is also encoded in $\mathcal{G}_{}$. Along the way, we prove that this algebra of primitive elements is also noetherian.
Keywords: Steenrod algebra, Unstable modules, Unstable algebras, Functors
Ouriel Blœdé 1
CC-BY 4.0
@article{AMBP_2025__32_2_185_0,
author = {Ouriel Bl{\oe}d\'e},
title = {$Nil$-closed {Noetherian} sub-algebras of $H^*(W)$ and their centres},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {185--220},
year = {2025},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {32},
number = {2},
doi = {10.5802/ambp.437},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.437/}
}
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%0 Journal Article %A Ouriel Blœdé %T $Nil$-closed Noetherian sub-algebras of $H^*(W)$ and their centres %J Annales mathématiques Blaise Pascal %D 2025 %P 185-220 %V 32 %N 2 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.437/ %R 10.5802/ambp.437 %G en %F AMBP_2025__32_2_185_0
Ouriel Blœdé. $Nil$-closed Noetherian sub-algebras of $H^*(W)$ and their centres. Annales mathématiques Blaise Pascal, Tome 32 (2025) no. 2, pp. 185-220. doi: 10.5802/ambp.437
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