Finite presheaves and ūĚíú-finite generation of unstable algebras mod nilpotents
Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2169-2204.

Inspired by the work of Henn, Lannes and Schwartz on unstable algebras over the Steenrod algebra modulo nilpotents, a characterization of unstable algebras that are ūĚíú-finitely generated up to nilpotents is given in terms of the associated presheaf. We do this by introducing the notion of a finite presheaf. In particular, this gives the natural characterization of the (co)analytic presheaves that are important in the theory of Henn, Lannes and Schwartz. An important source of examples is provided by unstable algebras of finite transcendence degree.

For unstable Hopf algebras, it is shown that the associated presheaf is finite if and only if its growth function is polynomial. This leads to a description of unstable Hopf algebras modulo nilpotents in the spirit of Henn, Lannes and Schwartz.

En s‚Äôinspirant du travail de Henn, Lannes et Schwartz sur la cat√©gorie des alg√®bres instables sur l‚Äôalg√®bre de Steenrod localis√©e loin des nilpotents, une caract√©risation des alg√®bres instables qui sont ūĚíú-finiment engendr√©es √† nilpotents pr√®s est donn√©e en termes du pr√©faisceau associ√© en utilisant la notion d‚Äôun pr√©faisceau fini, qui est introduite dans cet article. Ceci permet une caract√©risation naturelle des pr√©faisceaux (co)analytiques qui sont fondamentaux dans la th√©orie de Henn, Lannes et Schwartz. Une classe importante de pr√©faisceaux finis est fournie par les alg√®bres instables de degr√© de transcendance fini.

Pour les algèbres de Hopf instables, il est démontré que le préfaisceau associé est fini si et seulement si sa fonction de croissance est polynomiale. En particulier, ceci mène à une description de la catégorie des algèbres de Hopf, instables localisées loin des nilpotents.

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DOI: 10.5802/aif.3292
Classification: 55S10,  55U99
Keywords: unstable algebra, nilpotents, Steenrod algebra, finite generation, unstable Hopf algebra, presheaf, polynomial functor
Powell, Geoffrey 1

1 Laboratoire angevin de recherche en mathématiques (LAREMA), CNRS, Université d’Angers 2 bd Lavoisier 49045 Angers Cedex 01 (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Powell, Geoffrey. Finite presheaves and $\protect \mathscr{A}$-finite generation of unstable algebras mod nilpotents. Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2169-2204. doi : 10.5802/aif.3292. https://aif.centre-mersenne.org/articles/10.5802/aif.3292/

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