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Shachar Carmeli

Publications and source records attributed to Shachar Carmeli.

At least 19 recordsLinked to original sources
arXiv2026

Arithmetic Wu Formulas and the Generalized Hecke Theorem

We construct canonical Steenrod square operations on the Geisser--Schmidt/Milne modified compactly supported \'etale cohomology of separated finite-type schemes over rings of $S$-integers in which $2$ is invertible. This lets us extend Feng's notion of the absolute \'etale Wu class from the finite-field setting to arithmetic bases away from $2$. A key technical input is a modified compactly supported relative Wu formula, extending Benoist's relative Wu formula to the arithmetic compact-support setting. Using this, we prove an absolute Wu formula for regular projective flat schemes over either finite fields of odd characteristic or rings of $S$-integers away from $2$: if $f\colon X\to B$ is such a scheme, then the absolute Wu class of $X$ is the product of the relative Wu class $\operatorname{Sq}^{-1}(w_{\mathrm{et}}(\tau_f))$ and the pullback of the absolute Wu class of the base. In the $S$-integer case, the base contribution is $1+\beta_B$, where $\beta_B$ is the Bockstein, equivalently the Kummer class of $-1$. As an application, we obtain an infinite family of universal mod-$2$ congruences among the Chern classes of regular projective flat schemes over such bases, governed by an arithmetic deformation of Hirzebruch's $2$-Todd series; this is the generalized Hecke theorem. In low dimensions these congruences recover Hecke's theorem on the different away from $2$, Serre's Riemann--Hurwitz theorem for spin bundles, Atiyah's theorem on theta characteristics over finite fields, and the smooth $3$-manifold branched-cover analogue of the Shusterman--Sawin theorem, while yielding new higher-dimensional congruences over both finite and arithmetic bases.

math.NT↗
arXiv2025

Push-forward of smooth measures and strong Thom stratifications

We study the collection of measures obtained via push-forward along a map between smooth varieties over p-adic fields. We investigate when the stalks of this collection are finite-dimensional. We provide an algebro-geometric criterion ensuring this property. This criterion is formulated in terms of a canonical subvariety of the cotangent bundle of the source of the map.

math.AG↗
arXiv2025

Tate-valued Characteristic Classes II: Applications

We present a construction that manufactures $\E_\infty$ orientations of Tate fixed-point objects together with useful formulas for these maps, and then give a number of applications. For example, we produce a formula for the Frobenius homomorphisms of Thom spectra such as $\MU$ as well as certain lifts of Frobenius. We prove a rigidity property of $\MU$ as a \emph{cyclotomic} object. We construct a general obstruction theory for $\E_n$ complex orientations and establish various non-existence results for $p$-typical $\E_n$ orientations for low values of $p$ and $n$. We end with some miscellaneous further applications.

math.AT↗
arXiv2025

On the K-theory of algebraic tori

Given an algebraic torus $T$ over a field $F$, its lattice of characters $\Lambda$ gives rise to a topological torus $\mathfrak{T}(T)=\Lambda_{\mathbb R}/\Lambda$ with a continuous action of the absolute Galois group $G$. We construct a natural equivalence between the algebraic $K$-theory $K_{\ast}(T)$ and the equivariant homology $H^{G}_{\ast}(\mathfrak{T}(T);K_G(F))$ of the topological torus $\mathfrak{T}(T)$ with coefficients in the $G$-equivariant $K$-theory of $F$. This generalizes a computation of $K_0(T)$ due to Merkurjev and Panin. We obtain this equivalence by analyzing the motive $\mathbb{K}_{F}^{T}$ in the stable motivic category $\mathrm{SH}(F)$ of Voevodsky and Morel, where $\mathbb{K}_{F}$ is the motivic spectrum representing homotopy $K$-theory. We construct a natural comparison map $\mathfrak{F}\colon \mathbb{K}_{F}[B\Lambda] \to \mathbb{K}_{F}^{T}$ from the $\mathbb{K}_{F}$-homology of the \'etale delooping of $\Lambda$ to $\mathbb{K}_{F}^{T}$ as a special case of a motivic Fourier transform and prove that it is an equivalence by using a motivic Eilenberg--Moore formula for classifying spaces of tori.

math.KT↗
arXiv2025

Prismatic Steenrod operations and arithmetic duality on Brauer groups

We construct and analyze the "syntomic Steenrod algebra", which acts on the mod $p$ syntomic cohomology (also known as etale-motivic cohomology) of algebraic varieties in characteristic $p$. We then apply the resulting theory to resolve the last open cases of a 1966 Conjecture of Tate, concerning the existence of a symplectic form on the Brauer groups of smooth proper surfaces over finite fields. More generally, we exhibit symplectic structure on the higher Brauer groups of even dimensional varieties over finite fields. Although the applications are classical, our methods rely on recent advances in perfectoid geometry and prismatic cohomology, which we employ to define a theory of "spectral syntomic cohomology" with coefficients in motivic spectra. We then organize the resulting cohomology theories into a category of "spectral prismatic $F$-gauges", generalizing the prismatic $F$-gauges of Drinfeld and Bhatt--Lurie, for which we establish a ``spectral Serre duality'' extending classical coherent duality. These abstract constructions are leveraged to explicitly compute the syntomic Steenrod operations.

math.AG↗
arXiv2025

Tate-valued Characteristic Classes

We define a projective variant of classical complex orientation theory. Using this, we construct a map of spectra which lifts the total Chern class, providing an alternative answer to an old question of Segal \cite{segal}, previously answered by Lawson et al \cite{lawsonetal}. We also lift and generalize the ``sharp'' construction of Ando-French-Ganter \cite{afg} to an operation on arbitrary $\EE_\infty$-complex orientations, thereby providing a rich source of new $\EE_\infty$-orientations for commutative ring spectra. In particular we give an $\EE_\infty$-lift of the Jacobi orientation, a generalization of the much-studied two variable elliptic genus. Finally, we construct some new complex orientations of periodic ring spectra as requested in \cite{hahnyuan}.

math.AT↗
arXiv2024

The spectrum of units of algebraic $K$-theory

It is well known that the $[0,1]$ and $[0,2]$ Postnikov truncations of the units of the topological $K$-theories $\glone \KO$ and $\glone \KU$, respectively, are split, and that the splitting is provided by the ($\Z/2$-graded) line bundles. In this paper we give a similar splitting for the $[0,1]$-truncation of the units of algebraic $K$-theory, considered as a sheaf on affine schemes. A crucial step is to produce the splitting for $\glone K(\Z)$. Along the way we also give a complete calculation of the connective spectrum of strict units of $K(\Z)$ and $K(\F_\ell)$ for a prime $\ell$. Finally, we show that the units of algebraic $K$-theory do not split as a presheaf. In fact we show they do not even split pointwise.

math.KT↗
arXiv2024

Chromatic Cardinalities via Redshift

Using higher descent for chromatically localized algebraic $K$-theory, we show that the higher semiadditive cardinality of a $π$-finite $p$-space $A$ at the Lubin-Tate spectrum $E_n$ is equal to the higher semiadditive cardinality of the free loop space $LA$ at $E_{n-1}$. By induction, it is thus equal to the homotopy cardinality of the $n$-fold free loop space $L^n A$. We explain how this allows one to bypass the Ravenel-Wilson computation in the proof of the $\infty$-semiadditivity of the $T(n)$-local categories.

math.AT↗
arXiv2024

Maps between spherical group rings

We prove that for finitely generated abelian groups $A$ and $B$, the space of $\mathbb{E}_\infty$-ring maps between the spherical groups rings $\mathbb{S}[A] \to \mathbb{S}[B]$ is equivalent to the discrete set of group homomorphisms $A \to B$. We also prove generalizations where the sphere is replaced by other ring spectra, e.g. we give a formula for the strict units in group rings of the form $R[A]$ for $A$ a finite $p$-group and $R$ $p$-completely chromatically complete.

math.AT↗
arXiv2023

Descent and cyclotomic redshift for chromatically localized algebraic K-theory

We prove that $T(n+1)$-localized algebraic $K$-theory satisfies descent for $\pi$-finite $p$-group actions on stable $\infty$-categories of chromatic height up to $n$, extending a result of Clausen-Mathew-Naumann-Noel for finite $p$-groups. Using this, we show that it sends $T(n)$-local Galois extensions to $T(n+1)$-local Galois extensions. Furthermore, we show that it sends cyclotomic extensions of height $n$ to cyclotomic extensions of height $n+1$, extending a result of Bhatt-Clausen-Mathew for $n=0$. As a consequence, we deduce that $K(n+1)$-localized $K$-theory satisfies hyperdescent along the cyclotomic tower of any $T(n)$-local ring. Counterexamples to such cyclotomic hyperdescent for $T(n+1)$-localized $K$-theory were constructed by Burklund, Hahn, Levy and the third author, thereby disproving the telescope conjecture.

math.KT↗
arXiv2022

Higher semiadditive Grothendieck-Witt theory and the $K(1)$-local sphere

We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the $K(1)$-local sphere at the prime $2$. As a further application, we compute and clarify certain power operations in the homotopy of the $K(1)$-local sphere.

math.AT↗
arXiv2022

Characters and transfer maps via categorified traces

We develop a theory of generalized characters of local systems in $\infty$-categories, which extends classical character theory for group representations and, in particular, the induced character formula. A key aspect of our approach is that we utilize the interaction between traces and their categorifications. We apply this theory to reprove and refine various results on the composability of Becker-Gottlieb transfers, the Hochschild homology of Thom spectra, and the additivity of traces in stable $\infty$-categories.

math.AT↗
arXiv2022

The Chromatic Fourier Transform

We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height $n=0$, as well as a certain duality for the $E_n$-(co)homology of $π$-finite spectra, established by Hopkins and Lurie, at heights $n\ge 1$. We use this theory to generalize said duality in three different directions. First, we extend it from $\mathbb{Z}$-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of $E_n$. Second, we lift it to the telescopic setting by replacing $E_n$ with $T(n)$-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg--Moore formulas and Galois extensions in the telescopic setting. Third, we categorify their result into an equivalence of two symmetric monoidal $\infty$-categories of local systems of $K(n)$-local $E_n$-modules, and relate it to (semiadditive) redshift phenomena.

math.AT↗
arXiv2022

On the Strict Picard Spectrum of Commutative Ring Spectra

We compute the connective spectra of maps from $\mathbb{Z}$ to the Picard spectra of the spherical Witt vectors associated with perfect rings of characteristic $p$. As an application, we determine the connective spectrum of maps from $\mathbb{Z}$ to the Picard spectrum of the sphere spectrum.

math.AT↗
arXiv2021

$(A_\infty,2)$-categories and relative 2-operads

We define the notion of a 2-operad relative to an operad, and prove that the 2-associahedra form a 2-operad relative to the associahedra. Using this structure, we define the notions of an $(A_\infty,2)$-category and $(A_\infty,2)$-algebra in spaces and in chain complexes over a ring. Finally, we show that for any continuous map $A \to X$, we can associate an $(A_\infty,2)$-algebra $θ(A \to X)$ in $\textsf{Top}$, which specializes to $θ(\text{pt} \to X) = Ω^2 X$ and $θ(A \to \text{pt}) = ΩA \times ΩA$.

math.CT↗
arXiv2021

Relative de Rham Theory on Nash Manifolds

For a Nash submersion $ϕ\colon X\to Y$, we study the complex $\mathcal{SDR}(ϕ)$ of Schwartz sections of the relative de Rham complex of $ϕ$. We define the notion of Schwartz sections of constructible sheaves on Nash manifolds and prove that $\mathcal{SDR}(ϕ)$ is homotopy equivalent to the Schwartz sections of the proper push-forward $ϕ_!\mathbb{R}_X$ of the constant sheaf $\mathbb{R}_X$. Using this equivalence, we show that $\mathcal{SDR}(ϕ)$ depends (up to homotopy equivalence) only on the homology type of the map $ϕ$. We also deduce that $\mathcal{SDR}(ϕ)$ has Hausdorff homology spaces.

math.AG↗