A remark on Hopkins' chromatic splitting conjecture

The algebraic chromatic splitting conjecture for Noetherian ring spectra

We formulate a version of Hopkins' chromatic splitting conjecture for an arbitrary structured ring spectrum $R$, and prove it whenever $π_*R$ is Noetherian. As an application, these results provide a new local-to-global principle in the modular representation theory of finite groups.

Ambidexterity in Chromatic Homotopy Theory

We extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the $\infty$-categories of $T(n)$-local spectra are $\infty$-semiadditive for all $n$, where $T(n)$ is the telescope on a $v_{n}$-self map of a type $n$ spectrum. This extends and provides a new proof for the analogous result of Hopkins-Lurie on $K(n)$-local spectra. Moreover, we show that $K(n)$-local and $T(n)$-local spectra are respectively, the minimal and maximal $1$-semiadditive localizations of spectra with respect to a homotopy ring, and that all such localizations are in fact $\infty$-semiaddit…

String bordism and chromatic characteristics

We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)-local Hopkins-Miller classes $ζ_n$ take the places of the prime numbers, and this allows us to discuss higher bordism theories. We prove that the K(2)-localizations of the spectrum of topological modular forms as well as the string bordism spectrum have characteristic $ζ_2$.

An introduction to HKR character theory

The generalized character theory of Hopkins, Kuhn, and Ravenel is an important tool in the study of Morava E-theory and higher height phenomena in chromatic homotopy theory. In this paper, we provide an introduction to HKR character theory with many examples, applications, and alternative points of view.

Lubin-Tate theory, character theory, and power operations

This expository paper introduces several ideas in chromatic homotopy theory around Morava's extraordinary E-theories. In particular, we construct various moduli problems closely related to Lubin-Tate deformation theory and study their symmetries. These symmetries are then used in conjunction with the Hopkins-Kuhn-Ravenel character theory to provide formulas for the power operations and the stabilizer group action on the E-cohomology of a finite group.

Cofreeness in Real Bordism Theory and the Segal Conjecture

We prove that the genuine $C_{2^n}$-spectrum $N_{C_{2}}^{C_{2^n}}MU_{\mathbb{R}}$ is cofree, for all $n$. Our proof is a formal argument using chromatic hypercubes and the Slice Theorem of Hill, Hopkins, and Ravenel. We show that this gives a new proof of the Segal conjecture for $C_2$, independent of Lin's theorem.

Remarks on chromatically localised hermitian K-theory

We describe chromatic localisations of genuine L-spectra of discrete rings and deduce that the purity property of $K(1)$-local $K$-theory of rings established by Bhatt-Clausen-Mathew also holds in Grothendieck-Witt theory. In addition, we collect some results on higher chromatic localisations of various L-theory spectra and their consequences for Grothendieck-Witt theory.

等变稳定同伦理论中的周期现象

Periodic phenomena in equivariant stable homotopy theory

本文基于前人研究，构建了有限阿贝尔 p-群上的 A-等变色同伦理论框架，引入了 v_n-自映射与色塔的等变类比，并在 A = C_2 情况下将其与 Mahowald 不变量中的红移现象相联系。

色散同伦理论中的消失线

Vanishing lines in chromatic homotopy theory

本文证明了在素数2处Lubin-Tate谱的RO(G)分次同伦不动点谱序列存在强水平消失线并给出最佳界限，同时利用Hill-Hopkins-Ravenel范函子研究切片微分，确立了定向阶的界限。

通过不稳定同伦理论研究更高色散 Thom 谱

Higher chromatic Thom spectra via unstable homotopy theory

本文通过推广Hopkins-Mahowald定理，在T(n)、A和B上构造了BP⟨n-1⟩、ko和tmf等Thom谱，揭示了谱级别分裂与不稳定同伦理论中EHP序列的联系，并为幂零定理提供了新视角。

大素数下的色谱同伦是幺半代数性的

Chromatic Homotopy is Monoidally Algebraic at Large Primes

本文利用 operad 版 Goerss-Hopkins 阻碍理论，证明大素数条件下 $L_h$-局部谱的同伦 $(k,1)$-范畴具有对称幺半代数性，并给出 Lubin-Tate、KU 与 KO 模范畴的相应判据。

Tate cohomology and periodic localization of polynomial functors

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the univers…

Nilpotence in E_n Algebras

Nilpotence in the homotopy of $\mathbb{E}_\infty$-ring spectra is detected by the classical $H\mathbb{Z}$-Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in the homotopy of $\mathbb{E}_n$-ring spectra. For all odd primes $p$ and all chromatic heights $h$, we use the Cohen-Moore-Neisendorfer theorem to construct examples of $K(h)$-local, $\mathbb{E}_{2n-1}$-algebras with non-nilpotent $p^n$-torsion. We exploit the interaction of the Bousfield-Kuhn functor on odd spheres and Rezk's logarithm to show that our bound…