Des résultats d’annulation pour la tour de Taylor
[Results of cancellation for the Taylor tower]
Pham, Van Tuan1
1 Faculty of Mathematics-Mechanics-Informatics, Vietnam National University, Hanoi, 334 Nguyen Trai St., Thanh Xuan, Hanoi, Vietnam
Annales de l'Institut Fourier, Online first, 60 p.

We obtain vanishing results for derived functors in the sense of Dold–Puppe (Homologie nicht-additiver Funktoren. Anwendungen. Ann. Inst. Fourier Grenoble, 1961) and Taylor approximations of a functor according to Johnson–McCarthy (algebraic Goodwillie calculus) (Deriving calculus with cotriples. Trans. Amer. Math. Soc., 2004) using the combination of blocks in the category of strictly polynomial functors of Friedlander and Suslin (Cohomology of finite group schemes over a field. Invent. Math., 1997).

Nous obtenons des résultats d’annulation des foncteurs dérivés au sens de Dold–Puppe (Homologie nicht-additiver Funktoren. Anwendungen. Ann. Inst. Fourier Grenoble, 1961) et des approximations de Taylor d’un foncteur selon Johnson–McCarthy (calcul de Goodwillie algébrique) (Deriving calculus with cotriples. Trans. Amer. Math. Soc., 2004) en utilisant la combinatoire des blocs de la catégorie des foncteurs strictement polynomiaux de Friedlander et Suslin (Cohomology of finite group schemes over a field. Invent. Math., 1997).

Received:
Accepted:
Revised after acceptance:
Online First:
DOI: 10.5802/aif.3711
Classification: 18G30, 18G55, 20G05, 20G10
Mots-clés : Théorie des blocs, Foncteurs dérivé, Tour de Taylor, Foncteur strictement polynomial, Foncteur simple
Keywords: Block theory, Derived functor, Taylor tower, Strictly polynomial functors, simple functor

Pham, Van Tuan 1

1 Faculty of Mathematics-Mechanics-Informatics, Vietnam National University, Hanoi, 334 Nguyen Trai St., Thanh Xuan, Hanoi, Vietnam 
@unpublished{AIF_0__0_0_A145_0,
     author = {Pham, Van Tuan},
     title = {Des r\'esultats d{\textquoteright}annulation pour la tour de {Taylor}},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3711},
     language = {fr},
     note = {Online first},
}
TY  - UNPB
AU  - Pham, Van Tuan
TI  - Des résultats d’annulation pour la tour de Taylor
JO  - Annales de l'Institut Fourier
PY  - 2025
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3711
LA  - fr
ID  - AIF_0__0_0_A145_0
ER  - 
%0 Unpublished Work
%A Pham, Van Tuan
%T Des résultats d’annulation pour la tour de Taylor
%J Annales de l'Institut Fourier
%D 2025
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3711
%G fr
%F AIF_0__0_0_A145_0
Pham, Van Tuan. Des résultats d’annulation pour la tour de Taylor. Annales de l'Institut Fourier, Online first, 60 p.

[1] Akin, Kaan; Buchsbaum, David A. Characteristic-free representation theory of the general linear group, Adv. Math., Volume 58 (1985), pp. 149-200 | DOI | Zbl

[2] Akin, Kaan; Buchsbaum, David A. Characteristic-free representation theory of the general linear group. II. Homological considerations, Adv. Math., Volume 72 (1988) no. 2, pp. 171-210 | DOI | Zbl

[3] Akin, Kaan; Buchsbaum, David A.; Weyman, Jerzy Schur functors and Schur complexes, Adv. Math., Volume 44 (1982) no. 3, pp. 207-278 | DOI | Zbl

[4] Betley, Stanisław Stable derived functors, the Steenrod algebra and homological algebra in the category of functors, Fundam. Math., Volume 168 (2001) no. 3, pp. 279-293 | DOI | MR | Zbl

[5] Bott, Raoul On the Chern–Weil homomorphism and the continuous cohomology of Lie-groups, Adv. Math., Volume 11 (1973), pp. 289-303 | DOI

[6] Bousfield, A. K. Homogeneous functors and their derived functors, 1967 (mimeographed notes)

[7] Bousfield, A. K. Operations on derived functors of non-additive functors, 1967 (mimeographed notes)

[8] Bousfield, A. K.; Curtis, Edward B.; Kan, Daniel M.; Quillen, Daniel G.; Rector, David L.; Schlesinger, James W. The mod-p lower central series and the Adams spectral sequence, Topology, Volume 5 (1966), pp. 331-342 | DOI | MR | Zbl

[9] Breen, Lawrence; Mikhailov, Roman Derived functors of nonadditive functors and homotopy theory, Algebr. Geom. Topol., Volume 11 (2011) no. 1, pp. 327-415 | DOI | MR | Zbl

[10] Breen, Lawrence; Mikhailov, Roman; Touzé, Antoine Derived functors of the divided power functors, Geom. Topol., Volume 20 (2016) no. 1, pp. 257-352 | DOI | MR | Zbl

[11] Brown, Kenneth S. Cohomology of Groups, Graduate Texts in Mathematics, 182, Springer, 1982 | DOI | MR

[12] Séminaire Henri Cartan : Algèbres d’Eilenberg–MacLane et homotopie. 7e année 1954/55. Vols. I, II. Exposés I–XI, XII–XXII (Cartan, H.; Moore, J. C.; Thom, René; Serre, Jean-Pierre, eds.), Secrétariat Mathématique, 1955 | Zbl

[13] Cohen, Frederick R.; Hemmer, David J.; Nakano, Daniel K. On the cohomology of Young modules for the symmetric group, Adv. Math., Volume 224 (2010) no. 4, pp. 1419-1461 | DOI | MR | Zbl

[14] Curtis, Edward B. Lower central series of semi-simplicial complexes, Topology, Volume 2 (1963), pp. 159-171 | DOI | MR | Zbl

[15] Curtis, Edward B. Some relations between homotopy and homology, Ann. Math., Volume 82 (1965), pp. 386-413 | DOI | Zbl

[16] Dold, Albrecht; Puppe, Dieter Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier, Volume 11 (1961), pp. 201-312 | DOI | Numdam | MR | Zbl

[17] Donkin, Stephen On Schur algebras and related algebras. I, J. Algebra, Volume 104 (1986) no. 2, pp. 310-328 | DOI | Zbl

[18] Donkin, Stephen On Schur algebras and related algebras. II, J. Algebra, Volume 111 (1987) no. 2, pp. 354-364 | DOI | Zbl

[19] Eilenberg, Samuel; Mac Lane, Saunders On the groups H(Π,n). II. Methods of computation, Ann. Math., Volume 60 (1954), pp. 49-139 | DOI | Zbl

[20] Friedlander, Eric M.; Suslin, Andrei Cohomology of finite group schemes over a field, Invent. Math., Volume 127 (1997) no. 2, pp. 209-270 | DOI | MR | Zbl

[21] Green, James A. Polynomial representations of GL n , Lecture Notes in Mathematics, 830, Springer, 2007 | MR | Zbl

[22] James, Gordon; Kerber, Adalbert The representation theory of the symmetric group, Encyclopedia of Mathematics and Its Applications, 16, Addison-Wesley Publishing Group, 1981 (with an introduction by Gilbert de B. Robinson) | MR

[23] Jantzen, Jens Carsten Representations of algebraic groups, Pure and Applied Mathematics, 131, Academic Press Inc., 1987 | MR

[24] Jantzen, Jens Carsten Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, 2003 | MR

[25] Johnson, Brenda; McCarthy, Randy Linearization, Dold–Puppe stabilization and Mac Lane’s Q-construction, Trans. Am. Math. Soc., Volume 350 (1998) no. 4, pp. 1555-1593 | DOI | MR | Zbl

[26] Johnson, Brenda; McCarthy, Randy Trans. Amer. Math. Soc, Trans. Am. Math. Soc., Volume 356 (2004) no. 2, pp. 757-803

[27] Johnson, Brenda; McCarthy, Randy Taylor towers of symmetric and exterior powers, Fundam. Math., Volume 201 (2008) no. 3, pp. 197-216 | DOI | MR | Zbl

[28] Kantorovitz, Miriam Ruth; McCarthy, Randy The Taylor towers for rational algebraic K-theory and Hochschild homology, Homology Homotopy Appl., Volume 4 (2002) no. 1, pp. 191-212 | DOI | MR | Zbl

[29] Mac Lane, Saunders Homology, Classics in Mathematics, Springer, 1995 (reprint of the 1975 edition)

[30] Martin, Stuart Schur algebras and representation theory, Cambridge Tracts in Mathematics, 112, Cambridge University Press, 1993 | MR

[31] Pirashvili, Teimuraz I. Polynomial functors, Tr. Tbilis. Mat. Inst. Razmadze, Volume 91 (1988), pp. 55-66 | Zbl

[32] Pirashvili, Teimuraz I. Introduction to functor homology, Rational representations, the Steenrod algebra and functor homology (Panoramas et Synthèses), Volume 16, Société Mathématique de France, 2003, pp. 1-26 | Zbl

[33] Suslin, Andrei; Friedlander, Eric M.; Bendel, Christopher P. Infinitesimal 1-parameter subgroups and cohomology, J. Am. Math. Soc., Volume 10 (1997) no. 3, pp. 693-728 | DOI | Zbl

[34] Totaro, Burt Projective resolutions of representations of GL (n), J. Reine Angew. Math., Volume 482 (1997), pp. 1-13 | MR | Zbl

[35] Touzé, Antoine Cohomology of classical algebraic groups from the functorial viewpoint, Adv. Math., Volume 225 (2010) no. 1, pp. 33-68 | DOI | MR | Zbl

[36] Touzé, Antoine Ringel duality and derivatives of non-additive functors, J. Pure Appl. Algebra, Volume 217 (2013) no. 9, pp. 1642-1673 | DOI | MR | Zbl

[37] Touzé, Antoine Bar complexes and extensions of classical exponential functors, Ann. Inst. Fourier, Volume 64 (2014) no. 6, pp. 2563-2637 | DOI | Numdam | MR | Zbl

[38] Touzé, Antoine A functorial control of integral torsion in homology, Fundam. Math., Volume 237 (2017) no. 2, pp. 135-163 | DOI | MR | Zbl

[39] Touzé, Antoine On the structure of graded commutative exponential functors, Int. Math. Res. Not., Volume 2021 (2021) no. 17, pp. 13305-13415 | DOI | MR | Zbl

[40] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1994 | DOI | MR

[41] Weyman, Jerzy Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, 149, Cambridge University Press, 2003 | DOI | MR

Cited by Sources: