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The Tambara Structure of the Trace Ideal for Cyclic Extensions

Maxine Calle, Sam Ginnett, Harry Chen, Xinling Chen

Abstract

This paper explores the Tambara functor structure of the trace ideal of a Galois extension. In the case of a (pro-)cyclic extension, we are able to explicitly determine the generators of the ideal. Furthermore, we show that the absolute trace ideal of a cyclic group is strongly principal when viewed as an ideal of the Burnside Tambara Functor. Applying our results, we calculate the trace ideal for extensions of finite fields. The appendix determines a formula for the norm of a quadratic form over an arbitrary finite extension of a finite field.

The Tambara Structure of the Trace Ideal for Cyclic Extensions

Abstract

This paper explores the Tambara functor structure of the trace ideal of a Galois extension. In the case of a (pro-)cyclic extension, we are able to explicitly determine the generators of the ideal. Furthermore, we show that the absolute trace ideal of a cyclic group is strongly principal when viewed as an ideal of the Burnside Tambara Functor. Applying our results, we calculate the trace ideal for extensions of finite fields. The appendix determines a formula for the norm of a quadratic form over an arbitrary finite extension of a finite field.

Paper Structure

This paper contains 17 sections, 22 theorems, 79 equations.

Table of Contents

  1. Introduction and Background
  2. Background: Tambara, Burnside, Grothendieck, Witt, and Dress
  3. Tambara functors, ideals, and generators
  4. The Grothendieck-Witt and Burnside rings as Tambara functors
  5. The Dress map and the trace ideal
  6. The Burnside Tambara Functor for a Cyclic Group
  7. The Trace Ideal
  8. Odd degree extensions
  9. Cyclic 2-extensions
  10. General cyclic extensions
  11. The Trace Ideal for Profinite Extensions
  12. Tambara maps in the Burnside functor
  13. The trace ideal for profinite groups
  14. Some Applications and Examples
  15. Quadratic extensions
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose $\operatorname{Gal}(K/F) = C_N$ where $N$ has prime decomposition $2^\mu p_1^{\sigma_1}\cdots p_s^{\sigma_s}$. Then $\ker(\underline{A}_{C_N}\to \underline{GW}_F^K)$, seen as a Tambara ideal of $\underline{A}_{C_N}$, is generated by

Theorems & Definitions (44)

  • Theorem 1.1: see \ref{['thm:odd deg Cn extn']}, \ref{['thm:c2']}, \ref{['thm:c4']}, \ref{['thm:twopowers']}, and \ref{['thm:TI for Cn']}
  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.2
  • Definition 2.5
  • Remark 2.3
  • Definition 2.6: The Burnside ring
  • ...and 34 more