Variations on the Bloch-Ogus Theorem

A Bloch-Ogus Theorem for henselian local rings in mixed characteristic

We show a conditional exactness statement for the Nisnevich Gersten complex associated to an $\mathbb{A}^1$-invariant cohomology theory with Nisnevich descent for smooth schemes over a Dedekind ring with only infinite residue fields. As an application we derive a Nisnevich analogue of the Bloch-Ogus theorem for étale cohomology over a henselian discrete valuation ring with infinite residue field.

动机Bloch-Ogus理论中的可除现象

Divisibility phenomena in motivic Bloch--Ogus theory

本文研究了光滑项目ive变种上的动机Bloch-Ogus理论中的可除现象，证明了函数域的MilnorK群中无分歧类的子群包含在n可除元素子群中，并推广到任意双度数的动机上同调，进一步证明当k为有限或可分闭域且l为k中可逆素数时，X的动机上同调的Bloch-Ogus滤波中除了最后一步外都是l可除的，且推广到任意准项目ivek-方案。

Refined unramified cohomology of schemes

We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This recovers for cycles of low codimensions on smooth projective varieties previous results of Bloch--Ogus, Colliot-Thélène--Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch--Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel--Jacobi invariant has coniveau 1. This establishes a torsion version of a conjecture of Janns…

Moving lemmas in mixed characteristic and applications

The present paper contains new geometric theorems in mixed characteristic case. We derive a bunch of cohomological consequences using these geometric theorems. Among them an isotropy result for quadratic spaces, a purity result for quadratic spaces, a result on the Grothendieck--Serre conjecture for the special linear group of an Azumaya algebra. The Gersten conjecture for the functor K2 is proved. Bloch-Ogus type result is obtained as well. Suslin's exact sequence is derived and its application to a finiteness result is given. A version of the Roitman theorem is proved.

带支撑上同调的移动引理

A moving lemma for cohomology with support

对于一类自然的带支撑上同调理论（包括带适当系数的平展或pro-平展上同调），证明了在光滑拟射影k-簇（允许光滑射影紧化，如char(k)=0）上带支撑上同调类的移动引理，并由此得到Quillen、Bloch-Ogus和Gabber的消去定理的局部与全局推广、特征零的Gersten猜想的有限层次版本，以及平展上同调的单射性和余维1纯性定理的推广，进而表明[Sch23]中的精细非分支上同调群是motivic的。