Surjectivity of the \'Etale Excision Map for Homotopy Invariant Framed Presheaves

Andrei E. Druzhinin ; Ivan A. Panin

Informatics and Automation, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 103-127

Résumé

The categories of framed correspondences, framed presheaves, and framed sheaves were introduced by V. Voevodsky in his foundational “Notes on framed correspondences.” Based on these notes, G. Garkusha and I. Panin proposed a totally new approach to the stable motivic homotopy category $\mathrm {SH}(k)$. Their new description of the classical category $\mathrm {SH}(k)$ uses only local equivalences provided that $k$ is an infinite perfect field and the characteristic of $k$ is not $2$. The main aim of the present paper is to extend Garkusha and Panin's fundamental result on framed presheaves to all infinite perfect fields (including characteristic $2$). As a corollary, the local description of the category $\mathrm {SH}(k)$ is automatically valid without any restrictions on the characteristic of the base field. The heart of the present paper is the proof of the homotopy invariance of the Nisnevich sheaf $\mathcal F_{\mathrm{Nis}}$ associated to any homotopy invariant radditive quasi-stable framed presheaf $\mathcal F$ of abelian groups. Then, applying literally Garkusha and Panin's arguments, we deduce the strict homotopy invariance of the Nisnevich sheaf $\mathcal F_{\mathrm{Nis}}$.