Algebraically Closed Fields in Equivariant Algebra

TL;DR

This paper characterizes Nullstellensatzian $G$-Tambara functors as coinductions of algebraically closed fields, establishing a link between their $K$-theory spectra and those of algebraically closed fields, thus advancing the understanding of equivariant algebraic structures.

Contribution

It provides a characterization of Nullstellensatzian $G$-Tambara functors in terms of algebraically closed fields and relates their $K$-theory spectra, extending algebraic closure concepts to equivariant settings.

Findings

A $G$-Tambara functor is Nullstellensatzian iff it is the coinduction of an algebraically closed field.

Equivalence between the $K$-theory spectrum of Nullstellensatzian $G$-Tambara functors and algebraically closed fields.

Extension of algebraic closure concepts to equivariant algebra via Tambara functors.

Abstract

Using the Burklund-Schlank-Yuan abstraction of ``algebraically closed" to ``Nullstellensatzian", we show that a $G$-Tambara functor is Nullstellensatzian if and only if it is the coinduction of an algebraically closed field (for any finite group $G$). As a consequence we deduce an equivalence between the $K$-theory spectrum of any Nullstellensatzian $G$-Tambara functor with the $K$ theory of some algebraically closed field.