arXiv subjects
Publications and source records attributed to Tomer M. Schlank.
We prove an $\infty$-categorical version of the reflection theorem of Adámek-Rosický. Namely, that a full subcategory of a presentable $\infty$-category which is closed under limits and $κ$-filtered colimits is a presentable $\infty$-category. We then use this theorem in order to classify subcategories of a symmetric monoidal $\infty$-category which are equivalent to a category of modules over an idempotent algebra.
We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$, thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case $n=p=2$, we provide a full set of topological generators for these groups. Our arguments rely on recent advances in $p$-adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez--Dospinescu--Niziol.
We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion of the Witt vectors into the Lubin-Tate ring induces a split injection on continuous stabilizer cohomology with torsion cokernel of bounded exponent, thereby proving Hopkins' chromatic splitting conjecture and the vanishing conjecture of Beaudry-Goerss-Henn rationally. The key ingredients are the equivalence between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze-Weinstein, integral $p$-adic Hodge theory, and an integral refinement of a theorem of Tate on the Galois cohomology of non-archimedean fields.
We prove that $T(n+1)$-localized algebraic $K$-theory satisfies descent for $π$-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.
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.
At each prime $p$ and height $n+1 \ge 2$, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for $\mathbb{Z}$ acting by Adams operations on $\mathrm{BP}\langle n \rangle$, we prove that the $T(n+1)$-localized algebraic $K$-theory of $\mathrm{BP}\langle n \rangle^{h\mathbb{Z}}$ is not $K(n+1)$-local. We also show that Galois hyperdescent, $\mathbb{A}^1$-invariance, and nil-invariance fail for the $K(n+1)$-localized algebraic $K$-theory of $K(n)$-local $\mathbb{E}_{\infty}$-rings. In the case $n=1$ and $p \ge 7$ we make complete computations of $T(2)_*\mathrm{K}(R)$, for $R$ certain finite Galois extensions of the $K(1)$-local sphere. We show for $p\geq 5$ that the algebraic $K$-theory of the $K(1)$-local sphere is asymptotically $L_2^{f}$-local.
We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $K(n)$- and $T(n)$-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$, then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$, which are satisfied for example by the Lubin-Tate spectrum $E_n$, we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $T(n+1)$-localized K-theory, showing that they coincide for any $p$-invertible ring spectrum and for the completed Johnson-Wilson spectrum $\widehat{E(n)}$.
We construct Galois extensions of the T(n)-local sphere, lifting all finite abelian Galois extensions of the K(n)-local sphere. This is achieved by realizing them as higher semiadditive analogues of cyclotomic extensions. Combining this with a general form of Kummer theory, we lift certain elements from the K(n)-local Picard group to the T(n)-local Picard group.
Given a finite quandle $Q$, we study the average number of $Q$-colorings of the closure of a random braid in $B_n$ as $n$ varies. In particular we show that this number coincides with some polynomial $P_Q\in \mathbb{Q}[x]$ for $n\gg 0$. The degree of this polynomial is readily computed in terms of $Q$ as a quandle and these invariants are computed for all quandles with $|Q|\le 4$. Additionally we show that the methods in this paper allow to improve on the stability results of arXiv:0912.0325 from "periodic stability" to "stability".
Let $G$ be a reductive group and $X$ a smooth projective curve. We prove that, for $G$ classical and $σ$ an arbitrary $G$-local system on $X$, the space $\overline{\operatorname{Op}}^{gen}_{G,σ}$ of generic extended oper structures on $σ$ is homologically contractible. This contractibility result is crucial for the proof of the geometric Langlands conjecture.
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.
We show that Lubin--Tate theories attached to algebraically closed fields are characterized among $T(n)$-local $\mathbb{E}_{\infty}$-rings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$, the collection of $\mathbb{E}_\infty$-ring maps from $R$ to such Lubin-Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$ admits an $\mathbb{E}_\infty$-ring map to such a Lubin-Tate theory. As consequences, we construct $\mathbb{E}_{\infty}$ complex orientations of algebraically closed Lubin-Tate theories, compute the strict Picard spectra of such Lubin-Tate theories, and prove redshift for the algebraic $\mathrm{K}$-theory of arbitrary $\mathbb{E}_{\infty}$-rings.
The Segal conjecture describes stable maps between classifying spaces in terms of (virtual) bisets for the finite groups in question. Along these lines, we give an algebraic formula for the p-completion functor applied to stable maps between classifying spaces purely in terms of fusion data and Burnside modules.
In our previous paper, we constructed and studied a functorial extension of the evaluation map $S^1 \times \mathcal{L}X \to X$ to transfers along finite covers. In this paper, we show that this induces a natural evaluation map on the full subcategory of the homotopy category of spectra consisting of $p$-completed classifying spectra of finite groups. To do this, we leverage the close relationship between this full subcategory and the Burnside category of fusion systems.
We construct and study a functorial extension of the evaluation map $S^1 \times \mathcal{L} X \to X$ to transfers along finite covers. For finite covers of classifying spaces of finite groups, we provide algebraic formulas for this extension in terms of bisets. In the sequel, we show that this induces a natural evaluation map on the full subcategory of the homotopy category of spectra consisting of $p$-completed classifying spectra of finite groups.
Motivated by the philosophy that $C^*$-algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of $C^*$-algebras. We focus on $C^*$-algebras which are non-commutative CW-complexes in the sense of [ELP]. We construct the stable $\infty$-category of noncommutative CW-spectra, which we denote by $\mathtt{NSp}$. Let $\mathcal{M}$ be the full spectral subcategory of $\mathtt{NSp}$ spanned by "noncommutative suspension spectra" of matrix algebras. Our main result is that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on $\mathcal{M}$. To prove this we first prove a general result which states that any compactly generated stable $\infty$-category is naturally equivalent to the $\infty$-category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an $\infty$-categorical version of a result by Schwede and Shipley [ScSh1]. In proving this we use the language of enriched $\infty$-categories as developed by Hinich [Hin2,Hin3]. We end by presenting a "strict" model for $\mathcal{M}$. That is, we define a category $\mathcal{M}_s$ strictly enriched in a certain monoidal model category of spectra $\mathtt{Sp^M}$. We give a direct proof that the category of $\mathtt{Sp^M}$-enriched presheaves $\mathcal{M}_s^{op}\to\mathtt{Sp^M}$ with the projective model structure models $\mathtt{NSp}$ and conclude that $\mathcal{M}_s$ is a strict model for $\mathcal{M}$.
In a companion paper [ABS1] we introduced the stable $\infty$-category of noncommutative CW-spectra, which we denoted $\mathtt{NSp}$. Let $\mathcal{M}$ denote the full spectrally enriched subcategory of $\mathtt{NSp}$ whose objects are the non-commutative suspension spectra of matrix algebras. In [ABS1] we proved that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on $\mathcal{M}$. In this paper we investigate the structure of $\mathcal{M}$, and derive some consequences regarding the structure of $\mathtt{NSp}$. To begin with, we introduce a rank filtration of $\mathcal{M}$. We show that the mapping spectra of $\mathcal{M}$ map naturally to the connective $K$-theory spectrum $ku$, and that the rank filtration of $\mathcal{M}$ is a lift of the classical rank filtration of $ku$. We describe the subquotients of the rank filtration in terms of complexes of direct-sum decompositions which also arose in the study of $K$-theory and of Weiss's orthogonal calculus. We prove that the rank filtration stabilizes rationally after the first stage. Using this we give an explicit model of the rationalization of $\mathtt{NSp}$ as presheaves of rational spectra on the category of finite-dimensional Hilbert spaces and unitary transformations up to scaling. Our results also have consequences for the $p$-localization and the chromatic localization of $\mathcal{M}$.
We introduce and study the notion of \emph{semiadditive height} for higher semiadditive $\infty$-categories, which generalizes the chromatic height. We show that the higher semiadditive structure trivializes above the height and prove a form of the redshift principle, in which categorification increases the height by one. In the stable setting, we show that a higher semiadditive $\infty$-category decomposes into a product according to height, and relate the notion of height to semisimplicity properties of local systems. We place the study of higher semiadditivity and stability in the general framework of smashing localizations of $Pr^{L}$, which we call \emph{modes}. Using this theory, we introduce and study the universal stable $\infty$-semiadditive $\infty$-category of semiadditive height $n$, and give sufficient conditions for a stable $1$-semiadditive $\infty$-category to be $\infty$-semiadditive.