arXiv subjects

Noah Wisdom

Publications and source records attributed to Noah Wisdom.

11 recordsLinked to original sources

arXiv2026

The subgroup stratification of Nakaoka spectra

We construct a stratification on the Nakaoka spectrum of any $G$-Tambara functor indexed by the poset of subgroups of $G$. When $G$ is Dedekind, we show that the $H$th stratum of the Nakaoka spectrum of the Burnside $G$-Tambara functor is closed and not open; this provides examples of étale maps which induce closed, non-open maps on Nakaoka spectra. By computing the strata on the ghost of a $C_p$-Tambara functor we obtain many examples of étale maps of Tambara functors for which the induced map on Nakaoka spectra is not open, in contrast to the non-equivariant world. We also compute the strata of all fixed-point Tambara functors.

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arXiv2026

The $K$-theory of finite Tambara fields: away from $p$

In previous work, the author and David Chan computed the algebraic $K$-theory of the constant $C_2$-Tambara field with value the field with two elements, using a method which fails at odd primes. Herein we make progress towards the corresponding odd primary computations using a completely new idea. Particularly, we show that the $K$-theory groups of any constant $C_{p^n}$-Tambara field with value a characteristic $p$ finite field are torsion, and we completely determine these groups after inverting $p$. The away-from-$p$-torsion satisfies a simple pattern predicted by previous work, and a computer-aided computation shows that the $p$-power torsion is nontrivial in general.

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arXiv2026

Computations in Equivariant Topological Hochschild Homology

One of the most effective approaches to computations in algebraic $K$-theory is trace methods, which compare algebraic $K$-theory with topological Hochschild homology and topological cyclic homology. In recent work, two of the authors, together with Gerhardt, construct an equivariant refinement of topological Hochschild homology ($\mathrm{ETHH}$) which receives a trace map from Merling's genuine equivariant algebraic $K$-theory. In this paper, we perform foundational computations of $\mathrm{ETHH}$ that can serve as input for future computations of $\mathrm{ETHH}$ and equivariant topological cyclic homology. Namely, we compute $\mathrm{ETHH}(H\underline{\mathbb{F}}_p)$ for odd primes, showcasing the complexity of Bökstedt periodicity in this setting. Furthermore, we give computations of $\mathrm{ETHH}$ for the equivariant complex cobordism spectra $MU_G$ and $MU_{\mathbb{R}}$.

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arXiv2025

Affine étale group schemes over Tambara fields

We classify finite étale extensions and finite affine étale group schemes over the $G$-Tambara functor $\underline{\mathbb{F}}$, for $\mathbb{F}$ any algebraically closed field and $G$ any finite group. This establishes $G$-Galois descent from the Tambara functor algebraic closure of $\underline{\mathbb{F}}$. In particular, we find new families of étale extensions of any $G$-Tambara functor and show that, together with one of the families discovered by Lindenstrauss--Richter--Zou, these give all finite étale extensions of $\underline{\mathbb{F}}$. Our arguments also show that the map $\underline{K} \rightarrow \mathrm{FP}(L)$ associated to any $G$-Galois extension $L$ of $K$ is étale, generalizing a result of Lindenstrauss--Richter--Zou when $G$ is cyclic. Lastly, we classify flat finitely generated $\underline{\mathbb{F}}$-modules when $G = C_p$.

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arXiv2025

The algebraic $K$-theory of Green functors

In this paper we develop computational tools to study the higher algebraic $K$-theory of Green functors. We construct a spectral sequence converging to the algebraic $\mathbb{G}$-theory of any $G$-Green functor, for $G$ a cyclic $p$-group. From the spectral sequence we deduce a complete calculation of the algebraic $K$-theory of the constant $C_2$-Green functor associated to the field with two elements, and a calculation of the $p$-completion of the algebraic $K$-theory of the constant $G$-Green functor associated to the integers when $G$ is a cyclic $p$-group. Additionally, we introduce the notion of a Green meadow to abstract the Green functor structure underlying clarified Tambara fields, and show, under mild conditions, that every finitely generated projective module over a $G$-Green meadow is free when $G$ is a cyclic $p$-group. This gives a computation of $K_0$ for such Green functors.

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arXiv2025

Clarification and Coinduction of Tambara Functors

Tambara functors are equivariant analogues of rings arising in representation theory and equivariant homotopy theory. We introduce the notion of a clarified Tambara functor and show that under mild conditions every Tambara functor admits a decomposition as a product of coinductions of clarified Tambara functors; projection onto the non-coinduced part defines a reflective localization we call clarification. Through this perspective we study Morita invariance and $K$-theory of Tambara functors, field-like Tambara functors, and Nullstellensatzian clarified Tambara functors.

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arXiv2025

On minimal bases in homotopical combinatorics

We present a development in the computational suite for the study of $N_\infty$ operads for a finite group $G$. This progress is achieved using the simple yet powerful observation that Rubin's generation algorithm can be interpreted as a closure operator. Leveraging this perspective, we establish the existence of minimal bases for $N_\infty$ operads. By investigating these bases for certain families of groups we are led to introduce and analyze several novel combinatorial invariants for finite groups.

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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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arXiv2025

A classification of $C_{p^n}$-Tambara fields

Tambara functors arise in equivariant homotopy theory as the structure adherent to the homotopy groups of a coherently commutative equivariant ring spectrum. We show that if $k$ is a field-like $C_{p^n}$-Tambara functor, then $k$ is the coinduction of a field-like $C_{p^s}$-Tambara functor $\ell$ such that $\ell(C_{p^s}/e)$ is a field. If this field has characteristic other than $p$, we observe that $\ell$ must be a fixed-point Tambara functor, and if the characteristic is $p$, we determine all possible forms of $\ell$ through an analysis of the behavior of the Frobenius endomorphism and the trace of a $C_p$-Galois extension.

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arXiv2025

Real Global Group Laws and Hu-Kriz Maps

Recently, Hausmann defined global group laws and used them to prove that $MU^G_*$ is the $G$-equivariant Lazard ring, for $G$ a compact abelian Lie group. On the other hand, Hu and Kriz showed that the restriction map induces an isomorphism $M \mathbb{R}^{C_2}_{ρ*} \cong MU_{2*}$. In this paper, we blend these stories. We utilize the $C_2$-global spectrum $\mathbf{MR}$ defined by Schwede in an unpublished note, which gives rise to a genuine $G$-spectrum $M \mathbb{R}_η$ for each augmented compact Lie groups $η: G\to C_2$, simultaneously generalizing $MU_G$ and $M \mathbb{R}$. In the case of semi-direct product augmentations $G \rtimes C_2\to C_2$ with $G$ compact abelian Lie and $C_2$ acting by inversion, we show that the restriction along the inclusion $G \subset G \rtimes C_2$ is a split surjection $M \mathbb{R}^{G \rtimes C_2}_{ρ*} \rightarrow MU^{G}_{2*}$. Additionally, we propose an evenness conjecture, which implies that this map is an isomorphism. Along the way, we define Real $η$-orientations, Real global orientations, and corresponding notions of equivariant and global group laws.

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arXiv2024

Properties and Examples of $A$-Landweber Exact Spectra

It is classically known that Landweber exact homology theories (complex oriented theories which are completely determined by complex cobordism) admit no nontrivial phantom maps. Herein we propose a definition of $A$-Landweber exact spectra, for $A$ a compact abelian Lie group, and show that an analogous result on phantom maps holds. Also, we show that a conjecture of May on $KU_G$ is false. We do not prove an equivariant Landweber exact functor theorem, and therefore our result on phantom maps only applies to $MU_A$, $KU_A$, their $p$-localizations, and $BP_A$, which are shown to be $A$-Landweber exact by ad-hoc methods.

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