Notes on Cohomology

关于Tate上同调群有限性的一个注记

A note on finiteness of Tate cohomology groups

本文证明了对于定义在非阿基米德局部域上的约化代数群，在阶与剩余特征互素的素数阶自同构作用下，有限长表示的Tate上同调群仍是有限长表示，并探讨了其在通用性方面的应用。

Borel's stable range for the cohomology of arithmetic groups

In this note, we remark on the range in Borel's theorem on the stable cohomology of the arithmetic groups Sp(2n,Z) and SO(n,n;Z). This improves the range stated in Borel's original papers, an improvement that was known to Borel. Our main task is a technical computation involving the Weyl group action on roots and weights. This note originally appeared as the appendix to arXiv:1711.03139.

Notes on isocrystals

For varieties over a perfect field of characteristic p, etale cohomology with Q_l-coefficients is a Weil cohomology theory only when l is not equal to p; the corresponding role for l = p is played by Berthelot's rigid cohomology. In that theory, the coefficient objects analogous to lisse l-adic sheaves are the overconvergent F-isocrystals. This expository article is a brief user's guide for these objects, including some features shared with l-adic cohomology (purity, weights) and some features exclusive to the p-adic case (Newton polygons, convergence and overconvergence). The relationship bet…

On the Mod $p$ Cohomology of Pro-$p$ Iwahori Subgroups of $\operatorname{SL}_{n}(\mathbb Q_{p})$

This paper can be seen as an update to part of the author's dissertation. We study the mod $p$ cohomology of the pro-$p$ Iwahori subgroups $I$ of $\operatorname{SL}_{n}(\mathbb Q_{p})$ (and $\operatorname{GL}_{n}(\mathbb{Q}_{p})$) for $n=2$ and $n=3$. Here we use the spectral sequence $E_{1}^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_{p}) \Longrightarrow H^{s+t}(I,\mathbb{F}_{p})$ due to Sorensen, and we do explicit calculations with an ordered basis of $I$, which gives us a basis of $\mathfrak{g} = \mathbb{F}_{p} \otimes_{\mathbb{F}_{p}[π]} \operatorname{gr} I$ that we use to calculate $H^{s,t}(…

关于族中 étale $(\phi,\Gamma)$-模的一个注记

A note on étale $(φ,Γ)$-modules in families

本文利用形式概形棱镜位点上结构层的形变，建立棱镜 $(\Lambda,F)$-晶体与 $\Lambda$-étale 局部系统及其上同调的等价关系，并由此推出岩泽上同调同构、重证 Dee 的族等价定理。

A pamphlet on motivic cohomology

This short note is an "elementary'' introduction to the conjectural theory of motives.

On Beilinson's equivalence for $p$-adic cohomology

In this short note, we show a p-adic analogue of Beilinson's equivalence comparing two derived categories: the derived category of holonomic modules and derived category of modules whose cohomologies are holonomic.

A note on the Gauss-Manin connection for abelian schemes

We study differential forms on the universal vector extension $A^\natural$ of an abelian scheme $A$ in characteristic zero, and derive a new construction of the $D$-group scheme structure on $A^\natural$. This gives, in particular, a rather simple description of the Gauss--Manin connection on the de Rham cohomology of $A$ in terms of global algebraic differential forms on $A^\natural$. The key ingredient is the computation of the coherent cohomology of $A^{\natural}$, due to Coleman and Laumon.

Torsion in the crystalline cohomology of singular varieties

This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.

关于特征处重量滤化的注记

A note on weight filtrations at the characteristic

本文研究了在非可逆特征下，有理系数上射影Dedekind方案上的重量滤泡性质，并结合后续工作，证明了已知的对数上同调理论在项目ive sncd对上具有重量滤泡，同时展示了重量滤泡上同调是开部分的不变量。

Recovering function fields from their integral $\ell$-adic cohomology with the Galois action

In this note, we consider function fields of higher-dimensional algebraic varieties defined over non-local fields, and show how the Galois action on the cohomology such function fields can be used to parameterize their divisorial valuations. By applying a recent theorem of Pop, we observe that, in dimension $\geq 3$, this information is enough to completely determine the function field and the base-field in question.

非齐次Tate上链的杯积，以及分裂于循环扩张的局部域上环面的Galois上同调

Cup product of inhomogeneous Tate cochains, and Galois cohomology of tori over local fields that split over cyclic extensions

本文给出Tate上同调中杯积的非齐次上链公式，并用于计算分裂于循环扩张的局部域上环面的一阶上同调类的显式上循环。

上同调与同余

Cohomology and congruences

本文是2024年两所学校的课程笔记，系统介绍代数超曲面de Rham上同调中的p-adic方法，给出数论与组合学应用，并概述其在数学物理和算术几何中的意义。