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Algebraic homotopy classes of rational functions

Christophe Cazanave (2012)

Annales scientifiques de l'École Normale Supérieure

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.

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