Finite presheaves and đť’ś-finite generation of unstable algebras mod nilpotents
Annales de l'Institut Fourier, Volume 69 (2019) no. 5, p. 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 đť’ś-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’inspirant du travail de Henn, Lannes et Schwartz sur la catégorie des algèbres instables sur l’algèbre de Steenrod localisée loin des nilpotents, une caractérisation des algèbres instables qui sont 𝒜-finiment engendrées à nilpotents près est donnée en termes du préfaisceau associé en utilisant la notion d’un préfaisceau fini, qui est introduite dans cet article. Ceci permet une caractérisation naturelle des préfaisceaux (co)analytiques qui sont fondamentaux dans la théorie de Henn, Lannes et Schwartz. Une classe importante de préfaisceaux finis est fournie par les algèbres instables de degré 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.

Received : 2017-11-15
Revised : 2018-06-22
Accepted : 2018-09-25
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3292
Classification:  55S10,  55U99
Keywords: unstable algebra, nilpotents, Steenrod algebra, finite generation, unstable Hopf algebra, presheaf, polynomial functor
@article{AIF_2019__69_5_2169_0,
     author = {Powell, Geoffrey},
     title = {Finite presheaves and $\protect \mathscr{A}$-finite generation of unstable algebras mod nilpotents},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {5},
     year = {2019},
     pages = {2169-2204},
     doi = {10.5802/aif.3292},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_5_2169_0}
}
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/item/AIF_2019__69_5_2169_0/

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