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The Levelwise Finite Generation of Free Tambara Functors

Emory Sun

Abstract

We prove the levelwise finite generation of free polynomial $G$-Tambara functors in a collection of cases, most notably when $G$ is a finite Dedekind group or when $G \cong C_p \rtimes C_q$, $p > q$ primes. In the process, we establish the permanence of various finiteness conditions under box products and norms $n_H^G$ of Tambara functors, including a weak Hilbert Basis Theorem.

The Levelwise Finite Generation of Free Tambara Functors

Abstract

We prove the levelwise finite generation of free polynomial -Tambara functors in a collection of cases, most notably when is a finite Dedekind group or when , primes. In the process, we establish the permanence of various finiteness conditions under box products and norms of Tambara functors, including a weak Hilbert Basis Theorem.

Paper Structure

This paper contains 21 sections, 53 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Acknowledgements
  5. Review of Mackey and Tambara Functors
  6. The Polynomial Category
  7. Mackey and Tambara Functors
  8. Free Polynomial Tambara Functors
  9. Basic Properties of Free Tambara Functors
  10. A Levelwise Grading for Polynomials
  11. Finite Generation for Dedekind Groups
  12. Finiteness Results for General G
  13. Dedekind G, Transitive Case
  14. Finite Generation in Special Cases
  15. A Strong Constraint on G
  16. ...and 6 more sections

Key Result

Theorem A

Let $G$ be a finite group satisfying one of the following conditions: Then $G$ is Tambara finite.

Theorems & Definitions (121)

  • Definition : Definition \ref{['def:tambara_finite']}
  • Theorem A: Theorem \ref{['thm:finite_generation_general']}
  • Theorem B: Theorem \ref{['thm:relatively_finite_general']}
  • Theorem C: Weak Hilbert Basis Theorem, Theorem \ref{['thm:hilbert_basis']}
  • Theorem D: Theorem \ref{['thm:boxproduct']}
  • Theorem E
  • Theorem F: Corollary \ref{['cor:norm_preservation_general']}, Theorem \ref{['thm:norm_preservation']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 111 more