Nakaoka Spectrum in Tambara Theory

• Nakaoka Spectrum is the set of prime Tambara ideals in a G-Tambara functor, defined by multiplicative conditions analogous to classical prime ideals.
• Recent work employs a locale-theoretic framework via radical Tambara ideals, establishing a point-free, spatial, and coherent spectral topology.
• Studies reveal intricate phenomena such as nilpotence distinctions, subgroup stratifications, and explicit computational models in equivariant algebra.

Searching arXiv for papers on the Nakaoka spectrum and Tambara functors. The Nakaoka spectrum of a $G$-Tambara functor is the equivariant analogue of the Zariski spectrum: it is the set of prime Tambara ideals equipped with a Zariski-type topology (Chan et al., 2024). For a finite group $G$ and a $G$-Tambara functor $T$, Nakaoka’s definition takes prime ideals to be Tambara ideals $P \subsetneq T$ such that for all Tambara ideals $I,J \trianglelefteq T$, $IJ \subseteq P$ implies $I \subseteq P$ or $J \subseteq P$ (Chan et al., 2024). Recent work places this spectrum in a distinctly locale-theoretic framework: the frame $\mathop{RadId}_G(T)$ of radical Tambara ideals has points precisely the Nakaoka primes, is spatial and coherent, and recovers the topology of the Nakaoka spectrum, yielding a point-free proof that the space is spectral (Heard, 21 Apr 2026). Parallel developments analyze nilpotence, subgroup stratifications, explicit computations for fixed-point and ghost constructions, and extensions to bi-incomplete Tambara functors (Chan et al., 30 Jan 2026, Wisdom, 12 Aug 2025, Chan et al., 2024, Balchin et al., 8 May 2026).

1. Definition and basic topology

Let $G$0 be a finite group and $G$1 a $G$2-Tambara functor. A Tambara ideal is a family of ideals $G$3 closed under transfer, norm, restriction, and conjugation (Chan et al., 2024). A prime Tambara ideal is then defined by the multiplicative condition that for all Tambara ideals $G$4, if $G$5, then $G$6 or $G$7 (Chan et al., 2024). The Nakaoka spectrum is

$G$8

equipped with the Zariski topology whose closed sets are

$G$9

for Tambara ideals $G$0 (Chan et al., 2024). In the notation of the point-free construction, the corresponding basic opens may also be written

$G$1

(Heard, 21 Apr 2026).

When $G$2, Tambara functors are just commutative rings, and the Nakaoka spectrum recovers the usual Zariski spectrum (Chan et al., 2024). This places the theory directly in the lineage of classical commutative algebra, but with Tambara-specific phenomena arising from the interaction of restriction, transfer, and norm maps. One such phenomenon is that the product of Tambara ideals “may not be computed levelwise,” so primality is intrinsically global rather than a levelwise ring-theoretic condition (Chan et al., 30 Jan 2026).

2. Point-free construction via radical Tambara ideals

A point-free description is provided by the frame $G$3 of radical Tambara ideals (Heard, 21 Apr 2026). For a Tambara ideal $G$4, its radical is defined by

$G$5

where $G$6 denotes the Tambara ideal generated by $G$7 and products $G$8 are defined iteratively (Heard, 21 Apr 2026). A radical Tambara ideal is one satisfying $G$9 (Heard, 21 Apr 2026).

The poset of radical Tambara ideals, ordered by inclusion, forms a frame (Heard, 21 Apr 2026). Its meet is levelwise intersection, and its join is

$T$0

with $T$1 the levelwise sum of Tambara ideals (Heard, 21 Apr 2026). The frame-theoretic spectrum is defined by

$T$2

where $T$3 is the two-element frame, and the topology is generated by

$T$4

(Heard, 21 Apr 2026).

The main theorem identifies these points with the Nakaoka primes: $T$5 (Heard, 21 Apr 2026). The frame-theoretic opens $T$6 correspond to the basic open sets $T$7, and the topology generated by the $T$8 coincides with the Nakaoka topology defined by Nakaoka (Heard, 21 Apr 2026). In particular,

$T$9

so the frame of radical Tambara ideals recovers the frame of opens of the Nakaoka spectrum (Heard, 21 Apr 2026).

This approach closely parallels Joyal’s approach to the Zariski spectrum in commutative algebra (Heard, 21 Apr 2026). A plausible implication is that the locale-theoretic formulation isolates the topological content of Tambara ideal theory from any a priori reliance on points, while still recovering the usual prime spectrum once enough points are established.

3. Spectrality, coherence, and compact opens

A central structural result is that the Nakaoka spectrum is a spectral space (Heard, 21 Apr 2026, Chan et al., 30 Jan 2026). In the point-free account, the key inputs are that $P \subsetneq T$0 is spatial and coherent (Heard, 21 Apr 2026). Spatiality means the canonical map

$P \subsetneq T$1

is an isomorphism; for $P \subsetneq T$2, this holds by Theorem 5.6 (Heard, 21 Apr 2026). Coherence is established by identifying the compact elements as the radical finitely generated Tambara ideals,

$P \subsetneq T$3

for elements $P \subsetneq T$4 (Heard, 21 Apr 2026). Since a coherent frame is spatial and its point space is spectral, it follows that $P \subsetneq T$5 is spectral (Heard, 21 Apr 2026).

Chan and Spitz establish the same conclusion from the viewpoint of nilpotence: the Nakaoka spectrum of any Tambara functor is spectral, and the basic open sets $P \subsetneq T$6 are quasi-compact and form a basis (Chan et al., 30 Jan 2026). Their analysis also shows that $P \subsetneq T$7 is quasi-compact and sober (Chan et al., 30 Jan 2026). The point-free paper explicitly states that it recovers “a recent result of Chan and Spitz” (Heard, 21 Apr 2026).

The compact open subsets admit an explicit algebraic description. In the locale-theoretic treatment, the compact opens are exactly those

$P \subsetneq T$8

with $P \subsetneq T$9 a radical finitely generated Tambara ideal (Heard, 21 Apr 2026). This is the precise equivariant counterpart of the classical description of quasi-compact opens in the spectrum of a commutative ring.

4. Nilpotents, radicals, and localization phenomena

The relation between the Nakaoka spectrum and nilpotence is unusually delicate in equivariant algebra. Chan and Spitz prove that the nilradical of a Tambara functor $I,J \trianglelefteq T$0, defined as the intersection of all prime Tambara ideals, is computed levelwise: it consists precisely of the nilpotent elements in $I,J \trianglelefteq T$1 (Chan et al., 30 Jan 2026). Equivalently,

$I,J \trianglelefteq T$2

where $I,J \trianglelefteq T$3 is the nilradical of the commutative ring $I,J \trianglelefteq T$4 (Chan et al., 30 Jan 2026). They also prove that for a Tambara ideal $I,J \trianglelefteq T$5, the radical

$I,J \trianglelefteq T$6

is the intersection of all prime ideals containing $I,J \trianglelefteq T$7, and that $I,J \trianglelefteq T$8 is the radical of $I,J \trianglelefteq T$9 (Chan et al., 30 Jan 2026). This aligns with the radical used in the frame $IJ \subseteq P$0 (Heard, 21 Apr 2026).

At the same time, the papers emphasize a departure from ordinary commutative algebra. In Tambara functors, the nilpotents are not the same as the elements $IJ \subseteq P$1 such that $IJ \subseteq P$2 (Chan et al., 30 Jan 2026). Chan and Spitz call such elements kilpotent, and prove that

$IJ \subseteq P$3

is nilpotent in $IJ \subseteq P$4 (Chan et al., 30 Jan 2026). Every nilpotent is kilpotent, but not every kilpotent is nilpotent, and kilpotent elements do not in general form an ideal (Chan et al., 30 Jan 2026). Their Burnside functor example exhibits an element that is kilpotent but not nilpotent (Chan et al., 30 Jan 2026).

These results clarify why the topology of the Nakaoka spectrum can resemble classical algebraic geometry while still supporting genuinely equivariant phenomena. A plausible implication is that spectrum-theoretic arguments in Tambara functor theory must distinguish carefully between statements controlled by prime containment and statements controlled by localization.

5. Explicit descriptions and computational models

Several papers compute Nakaoka spectra in explicit families by reducing to ordinary commutative spectra. For the fixed point Tambara functor $IJ \subseteq P$5 of a $IJ \subseteq P$6-ring $IJ \subseteq P$7, there is a canonical homeomorphism

$IJ \subseteq P$8

(Chan et al., 2024). This identifies the Nakaoka spectrum with the GIT quotient of the classical Zariski spectrum (Chan et al., 2024). Relatedly, if $IJ \subseteq P$9 is a $I \subseteq P$0-Tambara functor with injective restrictions, then the $I \subseteq P$1th stratum of $I \subseteq P$2 is the set of prime ideals $I \subseteq P$3 such that $I \subseteq P$4 is a ring-theoretic prime ideal, and for fixed point functors one has a bijection with $I \subseteq P$5 (Wisdom, 12 Aug 2025).

For $I \subseteq P$6, the paper “On the Tambara Affine Line” introduces a ghost construction $I \subseteq P$7 for a $I \subseteq P$8-Tambara functor $I \subseteq P$9 (Chan et al., 2024). At the level of spectra,

$J \subseteq P$0

and the ghost map $J \subseteq P$1 is levelwise integral, so the induced map on spectra is surjective (Chan et al., 2024). This is used to compute the Nakaoka spectra of the complex representation ring Tambara functor $J \subseteq P$2 for $J \subseteq P$3, as well as the free Tambara functors on one generator (Chan et al., 2024).

For $J \subseteq P$4 over $J \subseteq P$5, the spectrum is described as

$J \subseteq P$6

(Chan et al., 2024). For the Tambara affine line over $J \subseteq P$7, the Nakaoka spectra of $J \subseteq P$8 and $J \subseteq P$9 are described in terms of the Zariski spectra of $\mathop{RadId}_G(T)$0, $\mathop{RadId}_G(T)$1, and $\mathop{RadId}_G(T)$2, assembled via explicit gluing (Chan et al., 2024). The same paper also establishes a weak Hilbert basis theorem, going up, lying over, and levelwise radicality of prime ideals in Tambara functors, and uses these results to compute Krull dimensions in examples (Chan et al., 2024).

6. Subgroup stratification and topological anomalies

A distinct structural refinement is the subgroup stratification of the Nakaoka spectrum (Wisdom, 12 Aug 2025). For each subgroup $\mathop{RadId}_G(T)$3, the $\mathop{RadId}_G(T)$4th stratum is defined as the image of the continuous map

$\mathop{RadId}_G(T)$5

(Wisdom, 12 Aug 2025). The stratification is indexed by the poset of subgroups of $\mathop{RadId}_G(T)$6, ordered by inclusion, and is natural in Tambara functor morphisms (Wisdom, 12 Aug 2025). This construction is motivated by comparisons with the Balmer spectrum in equivariant tensor-triangular geometry (Wisdom, 12 Aug 2025).

The behavior of these strata differs sharply from classical expectations. For the Burnside Tambara functor $\mathop{RadId}_G(T)$7, if $\mathop{RadId}_G(T)$8 is Dedekind, the $\mathop{RadId}_G(T)$9th stratum is

$G$00

hence closed and not open (Wisdom, 12 Aug 2025). The map $G$01 is étale, and the induced map on spectra is the inclusion of the $G$02th stratum; for proper $G$03, this image is closed but not open (Wisdom, 12 Aug 2025). The paper states this “in contrast to the non-equivariant world,” where étale maps induce open maps on spectra (Wisdom, 12 Aug 2025).

For ghosts of $G$04-Tambara functors, the $G$05th stratum can also fail to be open, yielding further examples of étale maps whose induced maps on Nakaoka spectra are not open (Wisdom, 12 Aug 2025). The paper also notes a “collision” phenomenon for $G$06, where points (p_{e,p