Algebraic homotopy classes of rational functions

Christophe Cazanave

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 4, page 511-534
  • ISSN: 0012-9593

Abstract

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Let  k be a field. We compute the set 𝐏 1 , 𝐏 1 N ofnaivehomotopy classes of pointed k -scheme endomorphisms of the projective line 𝐏 1 . Our result compares well with Morel’s computation in [11] of thegroup 𝐏 1 , 𝐏 1 𝐀 1 of  𝐀 1 -homotopy classes of pointed endomorphisms of  𝐏 1 : the set 𝐏 1 , 𝐏 1 N admits an a priori monoid structure such that the canonical map 𝐏 1 , 𝐏 1 N → 𝐏 1 , 𝐏 1 𝐀 1 is a group completion.

How to cite

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Cazanave, Christophe. "Algebraic homotopy classes of rational functions." Annales scientifiques de l'École Normale Supérieure 45.4 (2012): 511-534. <http://eudml.org/doc/272113>.

@article{Cazanave2012,
abstract = {Let $k$ be a field. We compute the set $\{\left[\mathbf \{P\}^1, \mathbf \{P\}^1 \right]\}^\{\mathrm \{N\}\}$ ofnaivehomotopy classes of pointed $k$-scheme endomorphisms of the projective line $\mathbf \{P\}^1$. Our result compares well with Morel’s computation in [11] of thegroup$\{\left[\mathbf \{P\}^1, \mathbf \{P\}^1 \right]\}^\{\mathbf \{A\}^1\}$ of $\{\mathbf \{A\}^1\}$-homotopy classes of pointed endomorphisms of $\mathbf \{P\}^1$: the set $\{\left[\mathbf \{P\}^1, \mathbf \{P\}^1\right]\}^\{\mathrm \{N\}\}$ admits an a priori monoid structure such that the canonical map $\{\left[\mathbf \{P\}^1, \mathbf \{P\}^1 \right]\}^\{\mathrm \{N\}\} \rightarrow \{\left[\mathbf \{P\}^1, \mathbf \{P\}^1 \right]\}^\{\mathbf \{A\}^1\}$ is a group completion.},
author = {Cazanave, Christophe},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {naive homotopy classes; rational functions; projective line; group completion},
language = {eng},
number = {4},
pages = {511-534},
publisher = {Société mathématique de France},
title = {Algebraic homotopy classes of rational functions},
url = {http://eudml.org/doc/272113},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Cazanave, Christophe
TI - Algebraic homotopy classes of rational functions
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 4
SP - 511
EP - 534
AB - Let $k$ be a field. We compute the set ${\left[\mathbf {P}^1, \mathbf {P}^1 \right]}^{\mathrm {N}}$ ofnaivehomotopy classes of pointed $k$-scheme endomorphisms of the projective line $\mathbf {P}^1$. Our result compares well with Morel’s computation in [11] of thegroup${\left[\mathbf {P}^1, \mathbf {P}^1 \right]}^{\mathbf {A}^1}$ of ${\mathbf {A}^1}$-homotopy classes of pointed endomorphisms of $\mathbf {P}^1$: the set ${\left[\mathbf {P}^1, \mathbf {P}^1\right]}^{\mathrm {N}}$ admits an a priori monoid structure such that the canonical map ${\left[\mathbf {P}^1, \mathbf {P}^1 \right]}^{\mathrm {N}} \rightarrow {\left[\mathbf {P}^1, \mathbf {P}^1 \right]}^{\mathbf {A}^1}$ is a group completion.
LA - eng
KW - naive homotopy classes; rational functions; projective line; group completion
UR - http://eudml.org/doc/272113
ER -

References

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  9. [9] F. Morel, Théorie homotopique des schémas, Astérisque 256 (1999). Zbl0933.55021
  10. [10] F. Morel, An introduction to 𝔸 1 -homotopy theory, in Contemporary developments in algebraic K -theory, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, 357–441 (electronic). Zbl1081.14029MR2175638
  11. [11] F. Morel, 𝐀 1 -algebraic topology over a field, Lecture Notes in Math. 2052, Springer, 2012. Zbl1263.14003MR2934577
  12. [12] F. Morel & V. Voevodsky, 𝐀 1 -homotopy theory of schemes, Publ. Math. I.H.É.S. 90 (1999), 45–143. Zbl0983.14007MR1813224
  13. [13] D. G. Quillen, Homotopical algebra, Lecture Notes in Math., No. 43, Springer, 1967. Zbl0168.20903MR223432

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