Algebraic homotopy classes of rational functions
[Classes d'homotopie algébrique de fractions rationnelles]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 4, pp. 511-534.

Soit k un corps. Nous déterminons l’ensemble 𝐏1,𝐏1N des classes d’homotopie naïve d’endomorphismes pointés de k-schémas de la droite projective 𝐏1. Notre résultat se compare bien avec le calcul de Morel [11] du groupe 𝐏1,𝐏1𝐀1 des classes d’𝐀1-homotopie d’endomorphismes pointés de 𝐏1 : l’ensemble 𝐏1,𝐏1N admet a priori une structure de monoïde pour laquelle l’application canonique 𝐏1,𝐏1N→𝐏1,𝐏1𝐀1 est une complétion en groupe.

Let k be a field. We compute the set 𝐏1,𝐏1N of naive homotopy classes of pointed k-scheme endomorphisms of the projective line 𝐏1. Our result compares well with Morel’s computation in [11] of the group 𝐏1,𝐏1𝐀1 of 𝐀1-homotopy classes of pointed endomorphisms of 𝐏1: the set 𝐏1,𝐏1N admits an a priori monoid structure such that the canonical map 𝐏1,𝐏1N→𝐏1,𝐏1𝐀1 is a group completion.

DOI : 10.24033/asens.2172
Classification : 14F42, 55Q40, 12Y05
Keywords: naive homotopy classes, rational functions, projective line, group completion
Mot clés : classes d'homotopie naïve, fractions rationnelles, droite projective, complétion en groupe 
@article{ASENS_2012_4_45_4_511_0,
     author = {Cazanave, Christophe},
     title = {Algebraic homotopy classes of rational functions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {511--534},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {4},
     year = {2012},
     doi = {10.24033/asens.2172},
     mrnumber = {3059240},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2172/}
}
TY  - JOUR
AU  - Cazanave, Christophe
TI  - Algebraic homotopy classes of rational functions
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2012
SP  - 511
EP  - 534
VL  - 45
IS  - 4
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/asens.2172/
DO  - 10.24033/asens.2172
LA  - en
ID  - ASENS_2012_4_45_4_511_0
ER  - 
%0 Journal Article
%A Cazanave, Christophe
%T Algebraic homotopy classes of rational functions
%J Annales scientifiques de l'École Normale Supérieure
%D 2012
%P 511-534
%V 45
%N 4
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/asens.2172/
%R 10.24033/asens.2172
%G en
%F ASENS_2012_4_45_4_511_0
Cazanave, Christophe. Algebraic homotopy classes of rational functions. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 4, pp. 511-534. doi : 10.24033/asens.2172. https://www.numdam.org/articles/10.24033/asens.2172/

[1] N. Bourbaki, Éléments de mathématique. Algèbre, chapitre IV: Polynômes et fractions rationnelles, Hermann, Paris, 1950 ; réédition Springer, 2007. | Zbl

[2] C. Cazanave, Classes d'homotopie de fractions rationnelles, C. R. Math. Acad. Sci. Paris 346 (2008), 129-133. | MR | Zbl

[3] C. Cazanave, Théorie homotopique des schémas d'Atiyah-Hitchin, thèse de doctorat, Université Paris 13, 2009.

[4] C. Cazanave, The 𝐀1-homotopy type of Atiyah-Hitchin schemes I: The geometry of complex points, preprint, 2010.

[5] I. M. GelʼFand, M. M. Kapranov & A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser, 1994. | MR | Zbl

[6] M. Karoubi & O. Villamayor, K-théorie algébrique et K-théorie topologique. I, Math. Scand. 28 (1971), 265-307. | MR | Zbl

[7] T. Y. Lam, Serre's problem on projective modules, Springer Monographs in Math., Springer, 2006. | MR | Zbl

[8] J. Milnor & D. Husemoller, Symmetric bilinear forms, Ergebn. Math. Grenzg. 73, Springer, 1973. | MR | Zbl

[9] F. Morel, Théorie homotopique des schémas, Astérisque 256 (1999). | Numdam | Zbl

[10] F. Morel, An introduction to 𝔸1-homotopy theory, in Contemporary developments in algebraic K-theory, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, 357-441 (electronic). | MR | Zbl

[11] F. Morel, 𝐀1-algebraic topology over a field, Lecture Notes in Math. 2052, Springer, 2012. | MR | Zbl

[12] F. Morel & V. Voevodsky, 𝐀1-homotopy theory of schemes, Publ. Math. I.H.É.S. 90 (1999), 45-143. | Numdam | MR | Zbl

[13] D. G. Quillen, Homotopical algebra, Lecture Notes in Math., No. 43, Springer, 1967. | MR | Zbl

  • Balch Barth, Viktor; Hornslien, William; Quick, Gereon; Wilson, Glen Matthew Making the motivic group structure on the endomorphisms of the projective line explicit, Advances in Mathematics, Volume 461 (2025), p. 110080 | DOI:10.1016/j.aim.2024.110080
  • Lerbet, Samuel Motivic stable cohomotopy and unimodular rows, Advances in Mathematics, Volume 436 (2024), p. 109415 | DOI:10.1016/j.aim.2023.109415
  • Bilu, Margaret; Ho, Wei; Srinivasan, Padmavathi; Vogt, Isabel; Wickelgren, Kirsten Quadratic enrichment of the logarithmic derivative of the zeta function, Transactions of the American Mathematical Society, Series B, Volume 11 (2024) no. 33, p. 1183 | DOI:10.1090/btran/201
  • Emprin, Coline; Horel, Geoffroy Weight structures and formality, arXiv (2024) | DOI:10.48550/arxiv.2406.19142 | arXiv:2406.19142
  • Brazelton, Thomas; McKean, Stephen; Pauli, Sabrina Bézoutians and the 𝔸1-degree, Algebra Number Theory, Volume 17 (2023) no. 11, p. 1985 | DOI:10.2140/ant.2023.17.1985
  • Asok, Aravind; Fasel, Jean; Hopkins, Michael J. Localization and nilpotent spaces in -homotopy theory, Compositio Mathematica, Volume 158 (2022) no. 3, p. 654 | DOI:10.1112/s0010437x22007321
  • Syed, Tariq Cancellation of vector bundles of rank 3 with trivial Chern classes on smooth affine fourfolds, Journal of Pure and Applied Algebra, Volume 226 (2022) no. 9, p. 107038 | DOI:10.1016/j.jpaa.2022.107038
  • Leo Kass, Jesse; Wickelgren, Kirsten An arithmetic count of the lines on a smooth cubic surface, Compositio Mathematica, Volume 157 (2021) no. 4, p. 677 | DOI:10.1112/s0010437x20007691
  • Brazelton, Thomas An Introduction to A1 -Enumerative Geometry, Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects, Volume 2292 (2021), p. 11 | DOI:10.1007/978-3-030-78977-0_2
  • Pauli, Sabrina; Wickelgren, Kirsten Applications to A1-enumerative geometry of the A1-degree, Research in the Mathematical Sciences, Volume 8 (2021) no. 2 | DOI:10.1007/s40687-021-00255-6
  • Kass, Jesse Leo; Wickelgren, Kirsten A classical proof that the algebraic homotopy class of a rational function is the residue pairing, Linear Algebra and its Applications, Volume 595 (2020), p. 157 | DOI:10.1016/j.laa.2019.12.041
  • Kass, Jesse Leo; Wickelgren, Kirsten The class of Eisenbud–Khimshiashvili–Levine is the local A1-Brouwer degree, Duke Mathematical Journal, Volume 168 (2019) no. 3 | DOI:10.1215/00127094-2018-0046
  • Asok, Aravind; Hoyois, Marc; Wendt, Matthias Affine representability results in 𝔸1–homotopy theory, II : Principal bundles and homogeneous spaces, Geometry Topology, Volume 22 (2018) no. 2, p. 1181 | DOI:10.2140/gt.2018.22.1181
  • Antieau, Benjamin; Elmanto, Elden A primer for unstable motivic homotopy theory, Surveys on Recent Developments in Algebraic Geometry, Volume 95 (2017), p. 305 | DOI:10.1090/pspum/095/01637
  • Horel, Geoffroy Motivic homological stability for configuration spaces of the line, Bulletin of the London Mathematical Society, Volume 48 (2016) no. 4, p. 601 | DOI:10.1112/blms/bdw032

Cité par 15 documents. Sources : Crossref, NASA ADS