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On the Tambara Affine Line

David Chan, David Mehrle, J. D. Quigley, Ben Spitz, Danika Van Niel

Abstract

Tambara functors are the analogue of commutative rings in equivariant algebra. Nakaoka defined ideals in Tambara functors, leading to the definition of the Nakaoka spectrum of prime ideals in a Tambara functor. In this work, we continue the study of the Nakoaka spectra of Tambara functors. We describe, in terms of the Zariski spectra of ordinary commutative rings, the Nakaoka spectra of many Tambara functors. In particular: we identify the Nakaoka spectrum of the fixed point Tambara functor of any $G$-ring with the GIT quotient of its classical Zariski spectrum; we describe the Nakaoka spectrum of the complex representation ring Tambara functor over a cyclic group of prime order $p$; we describe the affine line (the Nakaoka spectra of free Tambara functors on one generator) over a cyclic group of prime order $p$ in terms of the Zariski spectra of $\mathbb{Z}[x]$, $\mathbb{Z}[x,y]$, and the ring of cyclic polynomials $\mathbb{Z}[x_0,\ldots,x_{p-1}]^{C_p}$. To obtain these results, we introduce a "ghost construction" which produces an integral extension of any $C_p$-Tambara functor, the Nakaoka spectrum of which is describable. To relate the Nakaoka spectrum of a Tambara functor to that of its ghost, we prove several new results in equivariant commutative algebra, including a weak form of the Hilbert basis theorem, going up, lying over, and levelwise radicality of prime ideals in Tambara functors. These results also allow us to compute the Krull dimensions of many Tambara functors.

On the Tambara Affine Line

Abstract

Tambara functors are the analogue of commutative rings in equivariant algebra. Nakaoka defined ideals in Tambara functors, leading to the definition of the Nakaoka spectrum of prime ideals in a Tambara functor. In this work, we continue the study of the Nakoaka spectra of Tambara functors. We describe, in terms of the Zariski spectra of ordinary commutative rings, the Nakaoka spectra of many Tambara functors. In particular: we identify the Nakaoka spectrum of the fixed point Tambara functor of any -ring with the GIT quotient of its classical Zariski spectrum; we describe the Nakaoka spectrum of the complex representation ring Tambara functor over a cyclic group of prime order ; we describe the affine line (the Nakaoka spectra of free Tambara functors on one generator) over a cyclic group of prime order in terms of the Zariski spectra of , , and the ring of cyclic polynomials . To obtain these results, we introduce a "ghost construction" which produces an integral extension of any -Tambara functor, the Nakaoka spectrum of which is describable. To relate the Nakaoka spectrum of a Tambara functor to that of its ghost, we prove several new results in equivariant commutative algebra, including a weak form of the Hilbert basis theorem, going up, lying over, and levelwise radicality of prime ideals in Tambara functors. These results also allow us to compute the Krull dimensions of many Tambara functors.

Paper Structure

This paper contains 29 sections, 95 theorems, 198 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Computations
  3. Methods
  4. Notation
  5. Acknowledgements
  6. Prime Ideals of Tambara Functors
  7. The Polynomial Category and Tambara functors
  8. Tambara Ideals
  9. Prime Ideals
  10. The Zariski Topology
  11. A Weak Hilbert Basis Theorem
  12. Free Tambara Functors
  13. Noetherianity
  14. Tambara Primes are Levelwise Radical
  15. G-prime Ideals
  16. ...and 14 more sections

Key Result

Theorem A

Let $T$ be a Tambara functor.

Figures (2)

  • Figure 1: $\mathop{\mathrm{Spec}}\nolimits(\underline{\mathrm{A}})$ and $\mathop{\mathrm{Spec}}\nolimits( \text{ \normalfont }\text{ }(\underline{\mathrm{A}}))$
  • Figure 2: The action of the co-Tambara structure morphisms on the levelwise generators of the free $C_p$-Tambara functors.

Theorems & Definitions (203)

  • Theorem A: Topology of the Nakaoka spectrum, \ref{['SS:Topology']}
  • Theorem B: Nakaoka spectra as GIT quotients, \ref{['theorem: homeomorphisms of spec']}
  • Remark 1.1
  • Theorem C: The Tambara affine line over $C_p$, \ref{['SS:A1']}
  • Remark 1.3
  • Theorem D: \ref{['thm:RU']}
  • Theorem E: Going up, \ref{['theorem: going up']} and \ref{['cor: levelwise nil integral map gives surjection on spec']}
  • Theorem F: The ghost construction, \ref{['section: ghost']}
  • Theorem G: \ref{['theorem: krull dimension bounds for integral norm map']}
  • Definition 2.1
  • ...and 193 more