$K(n)$-local sphere is essentially the basic object of chromatic homotopy theory. However people even failed to completely understand its homotopy group so far. In 2024, Barthel, Schlank, Stapleton and Weinstein use input from $p$-adic geometry to fully understand the rational part of this homotopy group.

The key point is the fact that $L_{K(n)}\mathbb{S} \to E_n$ exhibts the latter as a pro-Galois extension of the former ring spectra with pro-Galois group $\mathbb G_n$ (the Morava stablizer group). In other words: $L_{K(n)}\mathbb {S}\simeq E_n^{h\mathbb G_n}$

The homotopy fixed point spectral sequence computing $L_{K(n)}\mathbb S\otimes \mathbb Q$ turns out to collapse immediately, hence there's no extra indeterminancy in this Galois descent process. In fact the only nontrivial terms in this spectral sequence is $H^*(\mathbb G_n, \pi_0 E_n)$.

The subtlety is that the action of the Morava stablizer group is in general hard to describe, this is where the p-adic geometry comes in.

Theorem.(Scholze-Weinstein) There exists a perfectoid space $\mathcal X$:

$LT_K\stackrel{GL_n(\Z_p)}{\longleftarrow} \mathcal X\stackrel{\mathcal O_D^\times}{\longrightarrow} \mathcal H_K$

where both arrows are pro-etale torsor, $LT_K$ denotes the Lubin-Tate space; $\mathcal H_K$ denotes (the base change of) the Drinfeld symmetric space.
Moreover, these two pro-etale torsor are dual to each other in the following sense: $\mathcal X$ carries an action of $GL_n(\Z_p)\times \mathcal O_D^\times$ action such that $GL_n(\Z_p)$-pro-etale torsor is $\mathcal O_D^\times$-equivariant and vice versa.

This duality phenomenon can be explained in the view point of "shtukas" and modification of vector bundles on Fargues-Fontaine curve. We'll provide a detailed proof of this theorem in terms of the language introduced by Scholze.

The previous result somehow translate the action of $\mathbb G_n$ (or $\mathcal O_D^\times$) on $\pi_0 E_n$ (i.e. the Lubin-Tate space) to the action of $GL_n(\Z_p)$ on the Drinfeld symmetric space $\mathcal H_K$. Considering these two torsors, intuitively one can translate the $\mathcal O_D^\times$-fixed point of the cohomology of $LT_K$ to the $GL_n(\Z_p)$-fixed point of the cohomology of $\mathcal H_K$.

Thus the remaining problems are the following two:

1. Precisely which cohomology theory can fit into the previous framework?
2. These cohomology might differ from the actual (global section of )structure sheaf, how to compare them?

The answer of the first question is that pro-etale cohomology and condensed solid group cohomology (fixed point) fits in well; the second question is the main computational part of the B-S-S-W's work.

To be precise, by constructing a comparison theorem one can show that the pro-etale cohomology $R\Gamma_{proet}$ and $\Gamma[\varepsilon]$ differs by bounded p-torsion, hence equivalence after rationalization. This comparison theorem is almost purely formal after a explicit computation for the case of a single point, i.e. $\mathrm{Spec}(K)$, whereas in this case the theorem turns out to be the integral refined version of the Ax-Sen-Tate theorem.

Notes

Rational K(n)-local sphere (incompleted)

This is a small seminar held in 2025 winter. The goal of this seminar is to understand how to construct the so called Bhatt-Morrow-Scholze filtration on topological cyclic homology (and its related variant) to extract the information of some usual p-adic cohomology theories.

The seminar is divided into the introductory part and three main parts. The introductory part introduce some backgrounds in p-adic Hodge theory and state some of the main results of Bhatt-Morrow-Scholze.

The first part mainly follows [BMS2], we briefly go through its main contents: constructing a so called $A_{\mathrm{inf}}$-cohomology theory by constructing a filtration on $\mathrm{THH},\mathrm{TC}^-$ and using some quasi-syntomic descent techniques. We also give the proof of all comparison theorems except for the $A_{\mathrm{inf}}$-etale comparison theorem.

The second part mainly follows [BS19], we study the complete version of $A_{\mathrm{inf}}$-cohomology, which is called prismatic cohomology. We introduce some basic properties and computational techniques such as Cech-Alexander complexes, prismatic coperfection etc. In this part we prove the crystalline comparison and Hodge-Tate comparison theorem. Moreover we use prisms to complete the proof of etale comparison theorem.

The third part mainly follows [Bha23], in this part we encode the cohomology theories we constructed before in a stack. To be precise, for any geometric object $X$, we associate a stack $X^C$ such that the cohomology of this stack encodes the $C$-cohomology theory of $X$. (For example one can take $C=HT,dR,$prismatic etc.) Moreover one can encode the additional structure of the cohomology theories, such as Hodge filtration, conjugate filtration (in pure characteeristic $p$) and Nygaard filtration. This is part is not discussed in the seminar but only appears in the seminar notes.

Main references (please check the seminar notes for detailed references):

• BMS2: Topological Hochschild homology and p-adic Hodge theory.
• NS18: On topological cyclic homology
• BS19: Prism and prismatic cohomology
• Kedlaya prismatic notes
• Bhatt prismatic lecture notes
• Bha23: Prismatic $F$-crystals.

Video link

Lecture 3: Outline of the construction, cotangent complex and quasisyntomic site
Lecture 4: $HC^-$ and de Rham complex, Hodge-Tate comparison for $A_{\mathrm{inf}}$
Lecture 5: Topological cohomology on perfectoid rings
Lecture 6: Crystalline-$A_{\mathrm{inf}}$ comparison, a glimpse on prismatic site
Lecture 7: Hodge-Tate comparison theorem for prismatic cohomology
Lecture 8: Prismatic coperfection, prismatic cohomology for quasiregular semiperfectoid rings, prismatic-$A_{\mathrm{inf}}$ comparison
Lecture 9: Prismatic-etale comparison theorem

Seminar notes

Complete seminar notes: BMSII

注记：Eichler-Shimura同构的相交上同调观点

个人认为讲稿中最有趣的部分在于Eichler-Shimura同构与偏屈层、相交上同调和Hodge结构的互动，很不幸地只有这一部分由于时间原因在讨论班上未能讲出，在此简述如下。

对于携带有划分$\mathcal S$的拓扑空间$X$，我们通常想象$X$为伪流形（粗略地说，由不同维数的流形粘合而成）此时其上携带者典范的划分$\mathcal S$。我们从现在固定这一背景设定。

在这一设定下，我们定义了$(X,\mathcal S)$上偏屈层和中间扩张(intermediate extension)的概念，前者是一类同调行为随着底空间维数变动的层，而后者描述了一个小空间$U\subseteq X,U\in\mathcal S$上的偏屈层$\mathcal L$如何尊重偏屈结构地推前，成为大空间上的层$j_{!*}\mathcal L$。

我们注意到偏屈层展现出的最重要的性质就是其同调行为尊重底空间的维数，但是观察Poincare对偶我们发现这恰好是一般空间（伪流形，复代数簇，...）上使得Poincare对偶成立所需要的关键性质。准确地说，使用六函子观点我们知道Poincare对偶无非说的是：

$\omega_X\coloneq p^!\mathbb Z,\quad p:X\to *$

满足$\omega_X$为定向层$\mathrm{O}$的平移$\mathrm{O}[n]$，特别地，当$X$可定向时$\omega_X=\Z[n]$。

更一般地，局部系的Poincare对偶无非说的是$p^!(-)=p^*(-)\otimes \omega_X$。因此上述讨论能够让人相信通过偏屈层我们能产出一个满足Poincare对偶的同调理论：这就是所谓相交上同调。

相交上同调. 给定复代数簇$X$以及光滑稠密开集$U,j:U\to X$，定义相交上同调为

$\mathrm{IH}^{n+i}(X,\mathcal L)\coloneq \mathbb H^i(X,j_{!*}\mathcal L[n])$

这里$\mathcal L$是$U$上局部系（自然是偏屈层：$U$光滑）。

偏屈层理论的精妙之处在于其还和Hodge结构有着紧密联系。粗略地说，任何“来自Hodge结构”的局部系产出的相交上同调都携带着Hodge结构：这是Saito等人的重要工作。我们简要解释一下：“来自Hodge结构”的正式名称是：Hodge结构变异(Variation of Hodge structure)。一个Hodge结构变异是指这个局部系$\mathcal V$（的函数层$\mathcal V\otimes_k \mathcal O_S$）上携带着一个下降滤过，满足其在每个纤维上都诱导了Hodge结构并满足所谓的Griffiths横截性：其上的平坦联络尊重滤过。

毫无疑问地，这样的结构在形变理论中随处可见：例如给定模空间$\mathcal M$，其对应的$p:\mathcal E\to \mathcal M$满足其在$m\in\mathcal M$处的纤维对应着$m\in\mathcal M$作为模空间中的点代表的结构，那么纤维的上同调上自然携带着Hodge结构（例如Kahler流形的形变），这些不同的Hodge结构跟随着纤维的形变一同形变，并粘合成为一模空间上的局部系。此时平坦联络为所谓的Gauss-Manin联络，这就自然给出了一个Hodge结构变异。

在Eichler-Shimura理论中我们考虑的正是椭圆曲线的模空间以及其上的万有（广义椭圆曲线），这恰好就是我们在上一段落中讨论的情形。更进一步，Eichler-Shimura同构在这个观点下恰好就是开模曲线$Y$上的局部系$R^1\pi_*\mathcal C=\pi_*\mathcal H^1_{dR}$诱导的相交上同调上的Hodge分解。

讲稿文件

Eichler-Shimura Theory

Descent in algebraic K-theory

主要参考资料为Weibel: The K-book和Hebeisteit: Lecture Notes for
Algebraic and Hermitian K-Theory

讲稿文件

Basics on algebraic K-theory

Plan

$K(n)$-局部球的有理同伦群计算。记录了使用的$p$-进几何背景以及一个利用Scholze-Berkeley给出的Lubin-Tate塔和Drinfeld塔之间的对偶（计算的主要input）。

Rational $K(n)$-local sphere (upcoming)

Seminar on Bhatt-Morrow-Scholze filtration (2024-2025 winter)

2025年寒假个人组织的一个小讨论班。

BMSII

Eichler-Shimura Theory

讨论班讲稿，简要介绍了Eichler-Shimura同构，以及权$k$模形式对应的Galois表示的构造细节。

Eichler-Shimura Theory

Descent in algebraic K-theory

讨论班讲稿，简要介绍了代数$K$理论（的局部化）满足的平展下降性质。

Descent in algebraic K-theory

Algebraic K Theory - Classical and Modern POV

关于代数K理论的基本内容，包含经典的构造和现代的观点。

Algebraic K Theory

Algebraic Topology and Homotopy Theory

这份笔记主要记录了代数拓扑和同伦论中学习到的内容，基于显然的原因以抄书为主。主要包含基础的代数拓扑、稳定同伦论和基本的色展同伦论语言。

Algebraic Topology

Complex Geometry

这份笔记的主要脉络源自经典教材：Griffiths&Harris - Principles of Algebraic Geometry，目前完成的内容涵盖了这本教材的Part 0 和 Part 1，包含了复几何的基础知识，同时这份笔记还记录了一些Akito Futaki教授在2024春开设的复几何课程中提到的额外内容。

Complex Geometry: Incompleted Version

Todo: Kodaira嵌入的证明，形变理论的PDE细节。
Plan. Voisin's textbooks on Hodge structure.

Commutative Algebra Notes based on Atiyah's textbook

参考书目: Atiyah - Introduction to Commutative Algebra.

注意：存在显著量的已知伪证。

Small Essays

Algebra Collection

一些代数上的小记录以及为了应付代数2课程而学习的简单Galois理论。
主要内容：

• $PSL(\mathbb F_q)$的单性
• Weddurbern小定理和Artin-Weddurbern定理
• Kan扩张
• 简单的Galois理论

Algebra Collection

Lecture Notes

Analysis 0H : Gu ChenLin. Sketch Version

Algebra 0H : Qiu Yu. Sketch Version

Algebraic Number Theory : Duan ZheFan. Sketch Version

Resources and Other Links

E-book download

Social Platform Accounts: Bilibili, Zhihu, NeteaseMusic

Recycle bin ♻️

These are rubbish.

Real Analysis

Reference Book: Royden Real Analysis 4th edition.

Incompleted Version

Small Essays

A quick review of Baby Rudin Chapter 7: Function Series

A summary of basic geometry I've learned.

(Originally intended to be written as notes for Baby Rudin Chapter 10; Caution: May contain factual errors since the author knew nothing about algebraic topology when writing this essay. )