Affine étale group schemes over Tambara fields

2025/08/12 by Wisdom, Noah · 1 citation
#14B25 (primary) 55P91 #14L15 (secondary) #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.09365

Abstract

We classify finite étale extensions and finite affine étale group schemes over the G-Tambara functor \underline\mathbbF, for \mathbbF any algebraically closed field and G any finite group. This establishes G-Galois descent from the Tambara functor algebraic closure of \underline\mathbbF. In particular, we find new families of étale extensions of any G-Tambara functor and show that, together with one of the families discovered by Lindenstrauss--Richter--Zou, these give all finite étale extensions of \underline\mathbbF. Our arguments also show that the map \underlineK → FP(L) associated to any G-Galois extension L of K is étale, generalizing a result of Lindenstrauss--Richter--Zou when G is cyclic. Lastly, we classify flat finitely generated \underline\mathbbF-modules when G = C_p.

Citations

• The algebraic K-theory of Green functors
• The subgroup stratification of Nakaoka spectra
• The Levelwise Finite Generation of Free Tambara Functors
• Clarification and Coinduction of Tambara Functors
• On the Tambara Affine Line
• Koszul Resolutions over Free Incomplete Tambara Functors for Cyclic p-Groups
• Examples of étale extensions of Green functors
• A C₂-equivariant analog of Mahowald's Thom spectrum theorem
• On the Andre-Quillen homology of Tambara functors

Cited by

• Inverting Integers in Tambara Functors

Related

• Examples of étale extensions of Green functors
• A classification of C_pⁿ-Tambara fields
• Étale algebras and the Kummer theory of finite Galois modules
• An étale realization which does not exist
• Tambara functors