标题： 积分动机上同调的$BSO_{4}$ 显示英文标题

标题： Integral Motivic Cohomology of $BSO_{4}$

摘要： 动机上同调是代数几何中的强大工具，其相关的实现映射提供了关于概形及其分类空间的上同调不变量之间关系的重要信息。计算这些分类空间的一般上同调不变量的问题仍在进行中。与本文最相关的是：(1) Totaro 对一般分类空间的 Chow 环的构造及其用于研究对称群的工作，见 arXiv:math/9802097，(2) Guillot 对李群$G_{2}$和$Spin(7)$的类似研究，见 arXiv:math/0508122，(3) Field 对$BSO(2n,\mathbb{C})$的 Chow 环的计算，见 arXiv:math/0411424，以及 (4) Yagita 对$\mathbb{Z}_{2}$-动机上同调的$BSO_{4}$和$BG_{2}$的研究，见 [Yag10]。本文中展示的工作涵盖了带有整系数的$BSO_{4}$的动机上同调的计算。主要方法借鉴了 Guillot 和 Yagita 提出的方法（arXiv:math/0508122，[Yag10]）。 这些结果为未来的工作奠定了基础，首先是对于$BG_{2}$的类似计算 ([Por21])。 显示英文摘要

摘要： Motivic cohomology is powerful tool in algebraic geometry with associated realization maps giving important information about the relations between cohomological invariants of schemes and their classifying spaces. The problem of computing general cohomological invariants of these classifying spaces is ongoing. Most relevant to this paper is (1) Totaro's construction of the Chow ring of a classifying space in general and his use of this to study symmetric groups in arXiv:math/9802097, (2) Guillot's similar examination for the Lie groups $G_{2}$ and $Spin(7)$ in arXiv:math/0508122, (3) Field's computation of the Chow ring of $BSO(2n,\mathbb{C})$ in arXiv:math/0411424, and (4) Yagita's work on the $\mathbb{Z}_{2}$-motivic cohomology of $BSO_{4}$ and $BG_{2}$ in [Yag10]. The work presented in this paper covers the computation of the motivic cohomology of $BSO_{4}$ with integral coefficients. The primary approach draws on methods laid out by Guillot and Yagita (arXiv:math/0508122, [Yag10]). These results lay the groundwork for future work, most immediately the analogous computation for $BG_{2}$ ([Por21]).