Morel's conjecture on the motivic homotopy sheaves of punctured affine space

Let n4n\geq 4 be an integer. Write Kn+2M\underline{\mathrm K}^{\mathrm M}_{n+2} for the unramified Milnor K-theory sheaf, πnA1(An{0})\operatorname{\underline \pi}_n^{\mathbb A^1}(\mathbb A^n-\{0\}) for the nnth A1\mathbb A^1-homotopy sheaf of punctured affine nn-space, and GWn+1[n]\underline{\operatorname{GW}}_{n+1}^{[n]} for the Nisnevich sheaf associated to the presheaf XGWn+1[n](X)X\mapsto \operatorname{GW}_{n+1}^{[n]}(X) of higher Hermitian K-theory. An exact sequence of unramified Nisnevich sheaves is conjectured:

Morel's conjecture.

0Kn+2M/24πnA1(An{0})GWn+1[n].0\longrightarrow \underline{\mathrm K}^{\mathrm M}_{n+2}/24\longrightarrow \operatorname{\underline \pi}_n^{\mathbb A^1}(\mathbb A^n-\{0\})\longrightarrow \underline{\operatorname{GW}}_{n+1}^{[n]}.

This conjecture describes the motivic homotopy sheaf of punctured affine space through Milnor K-theory and higher Hermitian K-theory. It has recently been settled in characteristic zero by Asok, Bachmann, and Hopkins.

Sources & referencesView supporting material

Primary source

Frédéric Déglise, “Motivic homotopy theory and stable homotopy groups”, arXiv:2510.17778 (2025).

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