On spectra and affine strict polynomial functors
Annales de l'Institut Fourier, Online first, 30 p.

We compare derived categories of the category of strict polynomial functors over a finite field and the category of ordinary endofunctors on the category of vector spaces. We introduce two intermediate categories: the category of -affine strict polynomial functors and the category of spectra of strict polynomial functors. They provide a conceptual framework for computational theorems of Franjou–Friedlander–Scorichenko–Suslin and clarify the role of inverting the Frobenius morphism in the comparison between rational and discrete cohomology.

Nous comparons des catégories dérivés de la catégorie des foncteurs strictement polynomiaux sur un domaine et la catégorie des endofoncteurs ordinaires sur la catégorie des espaces vectoriels. Nous introduisons deux catégories intermédiaires : la catégorie de foncteurs -affines strictement polynomiaux et la catégorie des spectres des foncteurs strictement polynomiaux. Ils fournissent un cadre conceptuel pour les théorèmes de calcul de Franjou–Friedlander–Scorichenko–Suslin et clarifient le rôle de l’inversion du morphisme de Frobenius dans la comparaison entre cohomologie rationnelle et discrète.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3666
Classification: 18A25, 18A40, 18G15
Keywords: Strict polynomial functor, Affine functor, Spectrum, Ext-group.
Mot clés : Foncteur strictement polynomial, foncteur affine, spectre, Ext-groupe.

Chałupnik, Marcin 1

1 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02–097 Warsaw (Poland)
@unpublished{AIF_0__0_0_A120_0,
     author = {Cha{\l}upnik, Marcin},
     title = {On spectra and affine strict polynomial functors},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3666},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Chałupnik, Marcin
TI  - On spectra and affine strict polynomial functors
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3666
LA  - en
ID  - AIF_0__0_0_A120_0
ER  - 
%0 Unpublished Work
%A Chałupnik, Marcin
%T On spectra and affine strict polynomial functors
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3666
%G en
%F AIF_0__0_0_A120_0
Chałupnik, Marcin. On spectra and affine strict polynomial functors. Annales de l'Institut Fourier, Online first, 30 p.

[1] Chałupnik, Marcin Schur-De Rham complex and its cohomology, J. Algebra, Volume 282 (2004) no. 2, pp. 699-727 | DOI | MR | Zbl

[2] Chałupnik, Marcin Derived Kan extension for strict polynomial functors, Int. Math. Res. Not. (2015) no. 20, pp. 10017-10040 | DOI | MR | Zbl

[3] Chałupnik, Marcin Affine strict polynomial functors and formality, Adv. Math., Volume 320 (2017), pp. 652-673 | DOI | MR | Zbl

[4] Djament, Aurélien; Touzé, Antoine Functor homology over an additive category (2022) (https://arxiv.org/pdf/2111.09719.pdf)

[5] Franjou, Vincent; Friedlander, Eric M.; Scorichenko, Alexander; Suslin, Andrei General linear and functor cohomology over finite fields, Ann. Math., Volume 150 (1999) no. 2, pp. 663-728 | DOI | MR | Zbl

[6] Franjou, Vincent; Lannes, Jean; Schwartz, Lionel Autour de la cohomologie de MacLane des corps finis, Invent. Math., Volume 115 (1994) no. 3, pp. 513-538 | DOI | MR | Zbl

[7] Franjou, Vincent; Pirashvili, Teimuraz Strict polynomial functors and coherent functors, Manuscr. Math., Volume 127 (2008) no. 1, pp. 23-53 | DOI | MR | Zbl

[8] Hovey, Mark Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra, Volume 165 (2001) no. 1, pp. 63-127 | DOI | MR | Zbl

[9] Hovey, Mark Model categories, Mathematical Surveys and Monographs, 63, American Mathematical Society, 2007 | DOI | Zbl

[10] Keller, Bernhard Deriving DG categories, Ann. Sci. Éc. Norm. Supér., Volume 27 (1994) no. 1, pp. 63-102 | DOI | Numdam | MR | Zbl

[11] Krause, Henning Localization theory for triangulated categories, Triangulated categories (London Mathematical Society Lecture Note Series), Volume 375, Cambridge University Press, 2010, pp. 161-235 | DOI | MR | Zbl

[12] Kuhn, Nicholas J. Generic representations of the finite general linear groups and the Steenrod algebra. III, K-Theory, Volume 9 (1995) no. 3, pp. 273-303 | DOI | MR | Zbl

[13] Kuhn, Nicholas J. A stratification of generic representation theory and generalized Schur algebras, K-Theory, Volume 26 (2002) no. 1, pp. 15-49 | DOI | MR | Zbl

[14] Touzé, Antoine Universal classes for algebraic groups, Duke Math. J., Volume 151 (2010) no. 2, pp. 219-249 | DOI | MR | Zbl

[15] Touzé, Antoine Troesch complexes and extensions of strict polynomial functors, Ann. Sci. Éc. Norm. Supér., Volume 45 (2012) no. 1, pp. 53-99 | DOI | Numdam | MR | Zbl

[16] Touzé, Antoine A construction of the universal classes for algebraic groups with the twisting spectral sequence, Transform. Groups, Volume 18 (2013) no. 2, pp. 539-556 | DOI | MR | Zbl

Cited by Sources: