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Ben Spitz

Publications and source records attributed to Ben Spitz.

10 recordsLinked to original sources
arXiv2026

Spectra of bi-incomplete Tambara functors

Bi-incomplete Tambara functors are equivariant generalizations of commutative rings. The most common forms of bi-incomplete Tambara functors are coefficient systems of commutative rings, Green functors, and Tambara functors. In the 1980s, Lewis introduced prime ideals in Green functors, and in the 2010s, Nakaoka introduced prime ideals in Tambara functors. In this work, we define the spectrum of prime ideals for an arbitrary bi-incomplete Tambara functor, simultaneously generalizing Lewis and Nakaoka's notions. We then produce many computational tools which we apply to several examples of interest.

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arXiv2026

Radicals and Nilpotents in Equivariant Algebra

Associated to each Tambara functor $T$ is its Nakaoka spectrum $\mathrm{Spec}(T)$, analogous to the Zariski spectrum of a commutative ring. We establish that this topological space is spectral. This result follows from an analysis of the notion of nilpotence in Tamabra functors. We prove that the nilradical of a Tambara functor $T$ (the intersection of all of its prime ideals) is computed levelwise, i.e. consists precisely of the nilpotent elements in $T$. In contrast to ordinary commutative algebra, the nilpotents of $T$ are not the same as the elements $x$ such that $T[1/x] = 0$; we therefore also give a classification of these elements. As a corollary, we observe that the set of these elements in $\pi_\star^s$ (the equivariant stable stems, viewed as an $\mathrm{RO}(G)$-graded Tambara functor) forms an ideal.

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arXiv2025

Inverting Integers in Tambara Functors

Let $G$ be a finite group, and $k$ an integer. In this note, we show that for any $G$-Tambara functor $T$ and any subgroups $H_1, H_2 \leq G$, $k$ is a unit in $T(G/H_1)$ if and only if $k$ is a unit in $T(G/H_2)$. In other words, one may speak unambiguously of the localization $T[1/k]$. As a consequence, the norm functors $N_H^G$ commute with inverting $k$.

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arXiv2025

$C_p$-Mackey functors in Macaulay2

We introduce the CpMackeyFunctors package for Macaulay2, which allows for computations with Mackey functors over a cyclic group of prime order.

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arXiv2025

Formal Concept Analysis and Homotopical Combinatorics

Formal Concept Analysis makes the fundamental observation that any finite lattice $(L, \leq)$ is determined up to isomorphism by the restriction of the relation ${\leq} \subseteq L \times L$ to the set $J(L) \times M(L)$, where $J(L)$ is the set of join-irreducible elements of $L$ and $M(L)$ is the set of meet-irreducible elements of $L$. For any finite lattice $L$ equipped with the action of a finite group $G$, we explicitly describe this restricted relation for the lattice of transfer systems $\mathsf{Tr}(L)$ in terms of $L$ only. We apply this to give new computations of the number of transfer systems for certain finite groups, and to produce bounds on the number of transfer systems on certain families of abelian finite groups. We also provide computer code to enable other researchers' use of these techniques.

math.CO↗
arXiv2025

The spectrum of the Burnside Tambara functor

We compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group. Our proof uses recent advances in the commutative algebra of Tambara functors, as well as a Tambara functor analogue of ghost coordinates which works over arbitrary finite groups and clarifies some previous computations. As examples, we explicitly compute the spectrum of the Burnside Tambara functor over all dihederal groups, the quaternion group $Q_8$, the alternating group $A_4$, and the general linear group $GL_3(F_2)$.

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arXiv2025

Algebraically Closed Fields in Equivariant Algebra

Using the Burklund-Schlank-Yuan abstraction of ``algebraically closed" to ``Nullstellensatzian", we show that a $G$-Tambara functor is Nullstellensatzian if and only if it is the coinduction of an algebraically closed field (for any finite group $G$). As a consequence we deduce an equivalence between the $K$-theory spectrum of any Nullstellensatzian $G$-Tambara functor with the $K$ theory of some algebraically closed field.

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arXiv2024

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.

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arXiv2024

Norms of Generalized Mackey and Tambara Functors

Let $G$ be a finite group. A $G$-Tambara functor can be defined as a product-preserving functor $\mathcal{P}_G \to \mathsf{Set}$ (satisfying one additional condition), where $\mathcal{P}_G$ is a category that is constructed in a straightforward way from the category of finite $G$-sets. By replacing the category of finite $G$-sets with other categories, we obtain a more general notion of "Tambara functor". This more general notion subsumes the notion of motivic Tambara functors introduced by Bachmann. In this article, we extend a result of Hoyer about $G$-Tambara functors to this more general context.

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