On the Tambara Affine Line

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.

💡 Research Summary

This paper develops a systematic algebraic‑geometric theory for Tambara functors, the equivariant analogue of commutative rings, by studying their prime‑ideal spectra as introduced by Nakaoka. The authors first establish basic topological properties of the Nakaoka spectrum: it is always quasi‑compact and sober, and when the Tambara functor is Noetherian the spectrum is Noetherian and closed subsets are precisely finite unions of points (Theorem A). These results parallel the classical Zariski spectrum and lay the groundwork for a “Tambara scheme” theory.

A central theme is the relationship between ordinary commutative rings with a finite group action and the associated fixed‑point Tambara functor FP(R). Theorem B shows that the Nakaoka spectrum of FP(R) is naturally homeomorphic to the GIT quotient Spec(R)//G, i.e. the spectrum of the invariant subring R^G. This bridges equivariant invariant theory and Tambara algebra, suggesting that many GIT quotients can be modeled by Tambara spectra.

The paper then focuses on explicit computations for the cyclic group Cₚ of prime order. The “Tambara affine line” A