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Algebraically Closed Fields in Equivariant Algebra

Jason Schuchardt, Ben Spitz, Noah Wisdom

TL;DR

The paper classifies Nullstellensatzian objects in the equivariant setting of $G$-Tambara functors, showing they are exactly coinductions of algebraically closed fields via $k \cong C_e^G(F)$. This yields a Morita equivalence between the module categories and, consequently, a canonical equivalence of K-theory spectra $K(k) \cong K(k(G/G))$, tying equivariant algebraic K-theory to the classical theory of algebraically closed fields. The approach blends BSY22’s Nullstellensatzian framework with Tambara-specific adjunctions, compactness arguments, and free-algebra technology, establishing a robust classification that extends to incomplete Tambara functors. Overall, the results illuminate the structure of field-like and Nullstellensatzian Tambara objects, provide concrete K-theoretic consequences, and suggest directions for equivariant Galois theory and chromatic phenomena in equivariant settings.

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.

Algebraically Closed Fields in Equivariant Algebra

TL;DR

The paper classifies Nullstellensatzian objects in the equivariant setting of -Tambara functors, showing they are exactly coinductions of algebraically closed fields via . This yields a Morita equivalence between the module categories and, consequently, a canonical equivalence of K-theory spectra , tying equivariant algebraic K-theory to the classical theory of algebraically closed fields. The approach blends BSY22’s Nullstellensatzian framework with Tambara-specific adjunctions, compactness arguments, and free-algebra technology, establishing a robust classification that extends to incomplete Tambara functors. Overall, the results illuminate the structure of field-like and Nullstellensatzian Tambara objects, provide concrete K-theoretic consequences, and suggest directions for equivariant Galois theory and chromatic phenomena in equivariant settings.

Abstract

Using the Burklund-Schlank-Yuan abstraction of ``algebraically closed" to ``Nullstellensatzian", we show that a -Tambara functor is Nullstellensatzian if and only if it is the coinduction of an algebraically closed field (for any finite group ). As a consequence we deduce an equivalence between the -theory spectrum of any Nullstellensatzian -Tambara functor with the theory of some algebraically closed field.
Paper Structure (17 sections, 64 theorems, 63 equations)

This paper contains 17 sections, 64 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Contents
  3. Notation and Terminology
  4. Acknowledgements
  5. Primer on Tambara Functors
  6. The Burnside Tambara functor and the free Green functor
  7. Coinduction
  8. Recollections on the Commutative Algebra of Tambara Functors
  9. Prime Ideals and Tambara Fields
  10. Free Algebras
  11. Adjunctions and Compact Algebras
  12. Nullstellensatzian Objects
  13. Sufficient Conditions for a Functor to Preserve Nullstellensatzian Objects
  14. Examples of Important Functors that Preserve Nullstellsatzian Objects
  15. Finitely Presented Fixed-Point Tambara Functors
  16. ...and 2 more sections

Key Result

Theorem A

Let $k$ be a Nullstellensatzian Tambara functor. Then $k$ is isomorphic to the coinduction of an algebraically closed field, i.e., $k$ is the fixed-point Tambara functor associated to the $G$-ring $\mathop{\mathrm{Fun}}\nolimits(G,F)$ for $F$ algebraically closed.

Theorems & Definitions (125)

  • Definition 1.1
  • Theorem A: cf. \ref{['theorem:alg-closed-iff-coinduced']}
  • Theorem B: cf. \ref{['cor:K-theory-equiv']}
  • Theorem C: cf. \ref{['thm:constant-preserves-compacts']}
  • Theorem D: cf. \ref{['prop:free-at-fixed-pts-is-levelwise-fg']}
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 115 more