On the study of cohomology of Chevalley groups using the etale cohomology thoery

用etale上同调理论研究Chevalley群的上同调

• 批准号: 12640025
• 负责人: MIMURA Mamoru
• 金额: $ 2.11万
• 依托单位: OKAYAMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640025/
• 关键词: Chevalley group; cohomology of group; spectral sequence; エタール・コホモロジー; 有限群; コホモロジー

The research project is to determine the cohomology of (the classifying space of) finite Chevalley groups, which consists of the following.(1) to construct, using the notion of algebraic geometry, the spectral sequence converging to the cohomology of (the classifying space of) finite Chevalley groups ;(2) to construct a complex giving the second term of the spectral sequence ;(3) to show the triviality of the spectral sequence.As for (1), we have succeeded in constructing the spectral sequence converging to the simplicial scheme which is the model of the Borel construction, by using the Deligne spectral sequence which is the algebraic version of the Eilenberg-Moore spectral sequence.Furthermore we use the Hochschild spectral sequence to show the triviality of the above spectral sequence, and thus we have succeeded in obtaining the spectral sequence mentioned in the above.As for (2), we have constructed concretely complexes giving the second term of the spectral sequence for all the cases of spinor type and of exceptional type.Finally, as for (3), we have some ideas to show the triviality of the spectral sequence by introducing some cohomology operations into the spectral sequence, which may need some more studies in the future.

本文的研究课题是确定有限Chevalley群（的分类空间）的上同调。(1)构造，使用代数几何的概念，谱序列收敛到的上同调有限Chevalley群（的分类空间）：（2）构造给出谱序列第二项的复形;（3）说明谱序列的平凡性，对于（1），我们成功地构造了收敛于单纯格式的谱序列，单纯格式是Borel构造的模型，利用Eilenberg-Moore谱序列的代数形式Deligne谱序列，进一步利用Hochschild谱序列证明了上述谱序列的平凡性，对于（2），我们构造了具体的复形，给出了所有旋量型和例外型情形的谱序列的第二项，最后，对于（3），我们有一些想法，通过在谱序列中引入一些上同调运算来说明谱序列的平凡性，这可能需要进一步的研究。

项目成果

K. Kuribayashi: "The cohomology of a pull-back on K-formal spaces"Topology and its Applications. (to appear). (2002)

K. Kuribayashi：“K-形式空间上的回拉的上同调”拓扑及其应用。

S.Ikenaga, S.Nitta, I.Yoshioka: "On the extensions of single valued continuous and set valued usc maps"Math. J. Okayama Univ.. 43. (2001)

S.Ikenaga、S.Nitta、I.Yoshioka：“关于单值连续和集值 USC 映射的扩展”数学。

S.Ikenaga, S.Nitta, I.Yoshioka: "On the extensions of single valued continuous and set valued usc maps"Math.J. Okayama Univ.. 43. (2001)

S.Ikenaga、S.Nitta、I.Yoshioka：“关于单值连续和集值 usc 映射的扩展”Math.J。

D.Buhagiar, I.Yoshioka: "Sums and products of ultracomplete topological spaces"Topology and its Applications. (2002)

D.Buhagiar、I.Yoshioka：“超完备拓扑空间的和与积”拓扑及其应用。

D. Buhagiar-I. Yoshioka: "Sums and products of ultracomplete topological spaces"Topology and its Applications. (to appear). (2002)

D.布哈吉亚尔-I。