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William Balderrama

Publications and source records attributed to William Balderrama.

At least 19 recordsLinked to original sources
arXiv2026

An algebraic model for rational ultracommutative rings

Given a global equivariant ultracommutative ring spectrum $E$ and inclusion $H\hookrightarrow G$ of finite groups, one may apply geometric fixed points to the norm $N_H^G E_H \to E_G$ to obtain what we call a \emph{geometric norm} $\Phi^H E \to \Phi^G E$. We prove that, together with inflations, these assemble into a functor $\Phi\colon\mathrm{UCom}_{\mathrm{fin}} \to \mathrm{Fun}(\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O}),\mathrm{CAlg})$, where $\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O})$ is the span category of finite connected groupoids with full backwards maps and faithful forwards maps, and that $\Phi$ restricts to an equivalence between full subcategories of rational objects. Central to our construction is a refinement of geometric fixed points to a natural transformation $\Phi\colon \mathrm{Sp}_\bullet\to\mathrm{Fun}(\mathrm{Orb}_\bullet^\simeq,\mathrm{Sp})$ which is compatible with restrictions and norms, and which restricts to an equivalence on full subcategories of rational objects. We explain how this may also be used to recover theorems of Barrero--Barthel--Pol--Strickland--Williamson and Wimmer on algebraic models for rational global spectra and normed $G$-commutative ring spectra respectively.

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arXiv2026

Kahn-Priddy theorems via the norm

We revisit the Kahn-Priddy theorem from the perspective of modern equivariant homotopy theory. This allows for a short proof that may be applied in other settings with sufficiently robust analogues of multiplicative norms and the Adams isomorphism. We illustrate this by establishing new Kahn-Priddy theorems in $L_n$ and $L_n^f$-local homotopy theory, motivic homotopy theory, and synthetic homotopy theory.

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arXiv2026

Ambidextrous global spectra and tempered cohomology

We introduce generalizations of global equivariant spectra which encode globally equivariant cohomology theories equipped with additional transfers, such as the deflation maps present in equivariant topological $K$-theory. We call these $\mathcal{Q}$-ambidextrous global spectra, where $\mathcal{Q}$ is a parameter encoding which additional transfers one allows. As our main example, we prove that the tempered cohomology theory associated with an oriented $\mathbf{P}$-divisible group, constructed by Lurie, is represented by a $π$-ambidextrous global $\mathbf{E}_\infty$ ring spectrum, encoding transfers along all relatively $π$-finite maps of global spaces. This is established by means of a general parametrized decategorification process, perhaps of independent interest, that produces $\mathcal{Q}$-ambidextrous global spectra from suitable global families of stable $\infty$-categories. By allowing $\mathcal{Q}$ to vary, we are able to coherently encode the fact that non-invertible morphisms of oriented $\mathbf{P}$-divisible groups induce maps of tempered theories that only commute with certain transfers. With these $π$-ambidextrous enhancements in hand, we explore the fundamental properties of tempered theories as equivariant stable homotopy types. We construct a well-behaved $F$-global homology theory for any $π$-finite space $F$, with good base change properties. Taking $F = \mathbf{B} H$ for a finite group $H$, this establishes general base change results for the geometric fixed points of tempered theories. We use this to compute the $H$-geometric fixed points of tempered theories, showing that they vanish for $H$ nonabelian and admit a simple algebro-geometric model when $H$ is abelian, with identifiable blueshift properties.

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arXiv2026

Unstable synthetic deformations I: Malcev theories

This paper is the first in a series of articles devoted to the construction and study of synthetic deformations of $\infty$-categories in the unstable context: that is, deformations of $\infty$-categories that categorify spectral sequence or obstruction-theoretic information. This paper sets up the foundations of our study. We introduce and study various classes of $\infty$-categorical and infinitary algebraic theories. We establish many basic properties of the $\infty$-categories of the models of different classes of theories, as well as recognition theorems identifying the $\infty$-categories that arise this way. We give an intrinsic definition of a Malcev theory in higher universal algebra. We establish that the $\infty$-category of models of a Malcev theory may be characterized as freely adjoining geometric realizations to the theory. This leads to the notion of a derived functor between $\infty$-categories of models of Malcev theories, and we study the behavior of these derived functors with respect to connectivity and limits. We recall the notion of a loop theory and study in detail the interaction between functors and derived functors of $\infty$-categories of loop models and models, establishing that a large class of comonads on the $\infty$-category of loop models deform canonically to the $\infty$-category of all models. In the last part of the paper, we show that by considering the coalgebras for these deformed comonads over $\infty$-categories of models, one can recover various stable deformations considered in the literature, such as filtered models or Postnikov-complete synthetic spectra. We then expand on these results by constructing $\infty$-categories of synthetic spaces and synthetic $\mathbf{E}_k$-rings.

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arXiv2026

Unstable synthetic deformations II: Infinitesimal extensions

This paper is the second in a series devoted to the study of unstable synthetic deformations through the lens of Malcev theories: certain $\infty$-categorical algebraic theories $\mathcal{P}$ with well-behaved $\infty$-categories $\mathrm{Model}_{\mathcal{P}}$ of models. In this paper, we show that Malcev theories and their models admit a well-behaved deformation theory, generalizing the classical deformation theory of rings and modules. As our main example, we prove that the Postnikov tower of a Malcev theory $\mathcal{P}$ is a tower of square-zero extensions, and that all of this structure is preserved by passage to $\infty$-categories of models. This allows us to control the difference between the $\infty$-categories $\mathrm{Model}_{h_{n+r}\mathcal{P}}$ and $\mathrm{Model}_{h_n\mathcal{P}}$ for $r \leq n$, and forms the basis of a ``cofibre of $τ$'' formalism in our approach to unstable synthetic homotopy theory. As an application, we derive from this a variety of new Blanc--Dwyer--Goerss style decompositions of moduli spaces of lifts along the tower $\mathrm{Model}_{\mathcal{P}}\to\cdots\to\mathrm{Model}_{h\mathcal{P}}$.

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arXiv2025

Affineness and reconstruction in complex-periodic geometry

Working in a generic derived algebro-geometric context, we lay the foundations for the general study of affineness and local descendability. When applied to $\mathbf{E}_\infty$ rings equipped with the fpqc topology, these foundations give an $\infty$-category of spectral stacks, a viable functor-of-points alternative to Lurie's approach to nonconnective spectral algebraic geometry. Specializing further to spectral stacks over the moduli stack of oriented formal groups, we use chromatic homotopy theory to obtain a large class of $0$-affine stacks, generalizing Mathew--Meier's famous $0$-affineness result. We introduce a spectral refinement of Hopkins' stack construction of an $\mathbf{E}_\infty$ ring, and study when it provides an inverse to the global sections of a spectral stack. We use this to show that a large class of stacks, which we call reconstructible, are naturally determined by their global sections, including moduli stacks of oriented formal groups of bounded height and the moduli stack of oriented elliptic curves.

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arXiv2025

Deformations of homotopy theories via algebraic theories

We develop a homotopical variant of the classic notion of an algebraic theory as a tool for producing deformations of homotopy theories. From this, we extract a framework for constructing and reasoning with obstruction theories and spectral sequences that compute homotopical data starting with purely algebraic data.

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arXiv2025

Type 2 complexes constructed from Brown-Gitler spectra

In a previous paper, one of us interpreted mod 2 Dyer-Lashof operations as explicit A-module extensions between Brown-Gitler modules, and showed these A-modules can be topologically realized by finite spectra occurring as fibers of maps between 2-local dual Brown-Gitler spectra. Starting from these constructions, in this paper, we show that infinite families of these finite spectra are of chromatic type 2, with mod 2 cohomology that is free over A(1). Applications include classifying the dual Brown-Gitler spectra after localization with respect to K-theory.

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arXiv2025

Equivariant $v_{1,\vec{0}}$-self maps

Let $G$ be a cyclic $p$-group or generalized quaternion group, $X\in \pi_0 S_G$ be a virtual $G$-set, and $V$ be a fixed point free complex $G$-representation. Under conditions depending on the sizes of $G$, $X$, and $V$, we construct a self map $v\colon\Sigma^V C(X)_{(p)}\rightarrow C(X)_{(p)}$ on the cofiber of $X$ which induces an equivalence in $G$-equivariant $K$-theory. These are transchromatic $v_{1,\vec{0}}$-self maps, in the sense that they are lifts of classical $v_1$-self maps for which the telescope $C(X)_{(p)}[v^{-1}]$ can have nonzero rational geometric fixed points.

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arXiv2024

$C_{p^n}$-equivariant Mahowald invariants

The classical Mahowald invariant is an operation that systematically produces new elements in the stable homotopy groups of spheres from known ones. We introduce the $C_{p^n}$-Mahowald invariant: a relation $\pi_\star S_{C_{p^{n-1}}} \rightharpoonup \pi_\ast S$ between the equivariant and classical stable stems which reduces to the classical Mahowald invariant when $n=1$. We compute the $C_{p^n}$-Mahowald invariants of all elements in the Burnside ring $A(C_{p^{n-1}}) = \pi_0 S_{C_{p^{n-1}}}$, extending Mahowald and Ravenel's computation of $M_{C_p}(p^k)$. As a consequence, we determine the image of the $C_p$-geometric fixed point map $\Phi^{C_p} : \pi_V S_{C_{p^n}} \to \pi_0 S_{C_{p^n}/C_p} \cong A(C_{p^{n-1}})$ when $V$ is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for $n=1$.

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arXiv2024

Equivalences of the form $Σ^V X \simeq Σ^W X$ in equivariant stable homotopy theory

We study equivalences of the form $Σ^{V}X\simeq Σ^{W}X$, where $G$ is a compact Lie group, $X$ is a $G$-spectrum, and $V$ and $W$ are $G$-representations. These equivalences encode a periodicity phenomenon in $G$-equivariant homotopy theory which generalizes the classical James periodicity for $G = C_2$. When $X = C(a_λ)$ is the cofiber of an Euler class, we construct an $RO(G)$-graded $J$-homomorphism $J\colon π_λKO_G\rightarrow π_\star^G C(a_λ)^\times$ which gives control over these periodicities. It also produces infinite periodic families in the $G$-equivariant stable stems. We illustrate this with several explicit examples. More generally, our work gives information about $RO(G)$-graded units in equivariant stable cohomotopy rings. We apply this to construct universal periodicities and differentials in the $G$-homotopy fixed point spectral sequence, and other equivariant Atiyah--Hirzebruch spectral sequences.

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arXiv2023

Total power operations in spectral sequences

We describe how power operations descend through homotopy limit spectral sequences. We apply this to describe how norms appear in the $C_2$-equivariant Adams spectral sequence, to compute norms on $π_0$ of the equivariant $KU$-local sphere, and to compute power operations for the $K(1)$-local sphere. An appendix contains material on equivariant Bousfield localizations which may be of independent interest.

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arXiv2023

Algebraic theories of power operations

We develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding well-behaved theories of power operations for $\mathbb{E}_\infty$ ring spectra. In particular, we consider Quillen cohomology in the context of algebras over algebraic theories, plethories, and Koszul resolutions for algebras over additive theories. By combining this general algebra with obstruction-theoretic machinery, we obtain tools for computing with $\mathbb{E}_\infty$ algebras over $\mathbb{F}_p$ and over Lubin-Tate spectra. As an application, we demonstrate the existence of $\mathbb{E}_\infty$ periodic complex orientations at heights $h\leq 2$.

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arXiv2023

A motivic analogue of the K(1)-local sphere spectrum

We identify the motivic $KGL/2$-local sphere as the fiber of $ψ^3-1$ on $(2,η)$-completed Hermitian $K$-theory, over any base scheme containing $1/2$. This is a motivic analogue of the classical resolution of the $K(1)$-local sphere, and extends to a description of the $KGL/2$-localization of an arbitrary motivic spectrum. Our proof relies on a novel conservativity argument that should be of broad utility in stable motivic homotopy theory.

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arXiv2023

The motivic lambda algebra and motivic Hopf invariant one problem

We investigate forms of the Hopf invariant one problem in motivic homotopy theory over arbitrary base fields of characteristic not equal to $2$. Maps of Hopf invariant one classically arise from unital products on spheres, and one consequence of our work is a classification of motivic spheres represented by smooth schemes admitting a unital product. The classical Hopf invariant one problem was resolved by Adams, following his introduction of the Adams spectral sequence. We introduce the motivic lambda algebra as a tool to carry out systematic computations in the motivic Adams spectral sequence. Using this, we compute the $E_2$-page of the $\mathbb{R}$-motivic Adams spectral sequence in filtrations $f \leq 3$. This universal case gives information over arbitrary base fields. We then study the $1$-line of the motivic Adams spectral sequence. We produce differentials $d_2(h_{a+1}) = (h_0+ρh_1)h_a^2$ over arbitrary base fields, which are motivic analogues of Adams' classical differentials. Unlike the classical case, the story does not end here, as the motivic $1$-line is significantly richer than the classical $1$-line. We determine all permanent cycles on the $\mathbb{R}$-motivic $1$-line, and explicitly compute differentials in the universal cases of the prime fields $\mathbb{F}_q$ and $\mathbb{Q}$, as well as $\mathbb{Q}_p$ and $\mathbb{R}$.

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arXiv2023

The Real-oriented cohomology of infinite stunted projective spaces

Let $E\mathbb{R}$ be an even-periodic Real Landweber exact $C_2$-spectrum, and $ER$ its spectrum of fixed points. We compute the $ER$-cohomology of the infinite stunted projective spectra $P_j$. These cohomology groups combine to form the $RO(C_2)$-graded coefficient ring of the $C_2$-spectrum $b(ER) = F(EC_{2+},i_\ast ER)$, which we show is related to $E\mathbb{R}$ by a cofiber sequence $Σ^σb(ER)\rightarrow b(ER)\rightarrow E\mathbb{R}$. We illustrate our description of $π_\star b(ER)$ with the computation of some $ER$-based Mahowald invariants.

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arXiv2023

An elementary proof of the chromatic Smith fixed point theorem

A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if A is a finite abelian p-group of rank r, then any finite A-space X which is acyclic in the nth Morava K-theory with n at least r will have its subspace F of fixed points acyclic in the (n-r)th Morava K-theory. This is a chromatic homotopy version of P.A.Smith's classical theorem that if X is acyclic in mod p homology, then so is F. The main purpose of this paper is to give an elementary proof of this new theorem that uses minimal background, and follows, as much as possible, the reasoning in standard proofs of the classical theorem. We also give a new fixed point theorem for finite dimensional, but possibly infinite, A-CW complexes, which suggests some open problems.

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