Motivic homotopy theory for perfect schemes

2025/10/01 by Dahlhausen, Christian, Hekking, Jeroen, Wolters, Storm
#14F42 #19E08 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2510.01390

Abstract

We construct a perfect version of Morel--Voevodsky's motivic homotopy category over a perfect base scheme in positive characteristic. By checking the axioms of a coefficient system, we establish a six-functor formalism. We show that multiplication by p is already invertible in the perfect motivic homotopy catgory. By work of Elmanto--Khan the functor sending an \mathbbF_p-scheme S to the category SH(S)[1/p] is invariant under universal homeomorphisms, hence under perfections. Our result gives an explicit model for the localization of SH at the universal homeomorphisms, which we conclude is the same as SH[1/p].

Citations

• Duality for KGL-modules in motivic homotopy theory
• Reconstruction of schemes from their étale topoi
• Norms in motivic homotopy theory
• Dimensional homotopy t-structure in motivic homotopy theory
• The six operations in equivariant motivic homotopy theory
• Projectivity of the Witt vector affine Grassmannian
• Affine Grassmannians and the geometric Satake in mixed characteristic
• Points in algebraic geometry
• Geometric models for higher Grothendieck-Witt groups in A1-homotopy theory
• Higher Topos Theory
• Spectra and symmetric spectra in general model categories

Related

• Norms in motivic homotopy theory
• Motivic homotopy theory of group scheme actions
• Localizations and completions in motivic homotopy theory
• Rational and p-local Motivic Homotopy Theory
• Stable motivic homotopy theory at infinity