Abstract
Let k be a field. Denote by \(\mathcal {S}\mathrm {pc}{}(k)_*\) the unstable, pointed motivic homotopy category and by \(R^{\mathbb A^1} \Omega _{\mathbb {G}_m}: \mathcal {S}\mathrm {pc}{}(k)_*\rightarrow \mathcal {S}\mathrm {pc}{}(k)_*\) the (\(\mathbb A^1\)-derived) \({\mathbb {G}_m}\)-loops functor. For a k-group G, denote by \(\mathrm {Gr}_{G}\) the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence \(R^{\mathbb A^1} \Omega _{\mathbb {G}_m}G \simeq \mathrm {Gr}_{G}\). We use this to compute the motive \(M(R^{\mathbb A^1} \Omega _{\mathbb {G}_m}G) \in \mathcal {DM}(k, \mathbb {Z}[1/e])\).
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Acknowledgements
I would like to thank Timo Richarz for patiently explaining many basic facts about affine Grassmannians, and in particular for explaining to me Lemma 13. I would also like to thank Maria Yakerson for comments on a draft, and Marc Hoyois for an enlightening discussion about the Grothendieck–Serre conjecture and the ldh topology. Finally I would like to thank an anonymous referee for suggesting a simplified exposition of Sect. 2.
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