AI 中文总结
该研究探讨光滑真代数簇的两类上同调在基域扩张下的性质，证明有限系数下弗罗贝尼乌斯纤维与基域无关，补充了特征处动机上同调的相关结论。
AI 中文摘要
我们研究光滑真代数簇$X$的动机上同调与平展动机上同调在代数闭基域$k$扩张下的变化情况。证明了在有限系数下，它们在特征外与$k$无关；在特征处，平展上同调确实依赖于$k$，而动机上同调仅在权$0$、$1$、$\text{dim}\thinspace X$时已知如此。接着考虑有限域上定义的代数簇（即韦伊-平展上同调）的动机上同调与平展动机上同调上的弗罗贝尼乌斯纤维，结合第一部分结果与完全幂幺群概形的结构理论，证明该纤维在有限系数下与$k$无关。最后给出整系数下的若干结果与例子。
英文摘要
We study how motivic and étale motivic cohomology of a smooth and proper variety $X$ changes under extensions of algebraically closed base fields $k$. We show that with finite coefficients, they are independent of $k$ away from the characteristic. At the characteristic, étale cohomology does depend on $k$, whereas for motivic cohomology this is only known in weights $0,1,\dim X$. We then consider the fiber of Frobenius on motivic cohomology and étale motivic cohomology for varieties defined over finite fields, i.e., Weil-étale cohomology. Combining the results of the first part with the structure theory of perfect unipotent group schemes, we show that this fiber is independent of $k$ with finite coefficients. Finally, we give some results and examples with integral coefficients.
