色高阶半可加性：通过红移

根的附加，代数 $K$-理论与色散红移

Adjunction of roots, algebraic $K$-theory and chromatic redshift

本文定义了 $E_1$-环的根附加和对数 THH，证明了根附加在代数 $K$-理论中诱导楔直和项的包含，并利用此结果证明了 Lubin-Tate 谱满足色散红移猜想。

Descent and vanishing in chromatic algebraic $K$-theory via group actions

We prove some $K$-theoretic descent results for finite group actions on stable $\infty$-categories, including the $p$-group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes's redshift philosophy: in particular, we show that if $R$ is an $\mathbb{E}_\infty$-ring spectrum with $L_{T(n)}R=0$, then $L_{T(n+1)}K(R)=0$. Our key observation is that descent and vanishing are logically interrelated, permitting to establish them simultaneously by induction on the height.

A higher chromatic analogue of the image of J

We prove a higher chromatic analogue of Snaith's theorem which identifies the K-theory spectrum as the localisation of the suspension spectrum of CP^\infty away from the Bott class; in this result, higher Eilenberg-MacLane spaces play the role of CP^\infty = K(Z,2). Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the K(n)-local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross-Hopkins dual of the sphere. Our main technical tool is a K(n)-local notion generalising complex orientation to higher Eilenberg-MacLane spaces.…

The Chromatic Nullstellensatz

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}_{\inf…

Redshift and multiplication for truncated Brown-Peterson spectra

We equip $\mathrm{BP} \langle n \rangle$ with an $\mathbb{E}_3$-$\mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map $\mathrm{K}(\mathrm{BP}\langle n \rangle)_{(p)} \to \mathrm{L}_{n+1}^{f} \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}$ has bounded above fiber.

关于更高拓扑Hochschild同调的注记

A note on higher topological Hochschild homology

本文通过更高拓扑Hochschild同调的同调固定点谱研究更高色红移现象，揭示了检测更高$ v_n $-元素的可能性。

色移

Chromatic Noshift

本文研究了在刚性对称单状稳定∞-范畴中，代数K-理论是否提升色高，发现其失败并构造了反例，同时推广了Levy关于环的范畴化的猜想。

Quillen-Lichtenbaum 导出的红移边界

The Redshift Bound from Quillen-Lichtenbaum

该研究针对代数K理论的色高度提升问题，基于Lubin-Tate谱的下降方法，借助Hahn-Wilson的Quillen-Lichtenbaum结果给出了Clausen等人结论的新证明。

有效红移

Effective Redshift

本文针对Quillen-Lichtenbaum猜想的有效版本，提出了基于syntomic上同调的简单准则以确定局部化映射为同构的度数，并结合近期计算证明了在BP<n>、k(n)和ko等环谱情况下有限局部化映射是(-1)-截断的。

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…