Motivic Steenrod operations at the characteristic via infinite ramification

2025/06/05 by Toni Annala, Elden Elmanto, Annala, Toni +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.05585

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct motivic power operations on the mod-p motivic cohomology of \Fb_p-schemes using a motivic refinement of Nizioł's theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations satisfy the expected properties (naturality, Adem relations, and the Cartan formula) for all bidegrees, generalizing previous results of Primozic which were only know along the ``Chow diagonal.'' We offer geometric applications of our construction: 1) an example of non-(quasi-)smoothable algebraic cycle at the characteristic, 2) an answer to the motivic Steenrod problem at the characteristic, 3) a counterexample to the integral version of a crystalline Tate conjecture.

Related

• Motivic Steenrod operations in characteristic p
• The motivic Steenrod algebra in positive characteristic
• Reduced power operations in motivic cohomology
• Opérations de Steenrod motiviques
• On extension of the motivic cohomology beyond smooth schemes