The cohomslogy group of the classifying space

分类空间的上同调群

• 批准号: 13640006
• 负责人: YAGITA N
• 金额: $ 1.02万
• 依托单位: IBARAKI UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640006/
• 关键词: Cohomology group; Classifying space; BP-theory; Chow ring

For a smooth complex algebraic variety X, the group CH^I(X) of codimension I algebraic cycles modulo rational equivalence assemble to the Chow ring CH^*(X) = Σ_iCH^I(X). Totaro constructed a map c^^~l : CH^*(X) → BP^*(X) 【cross product】_<BP>・ Z_<(p)> such that the compositioncl : CH^I(X)_<(p)> →^^<c^^~l> BP^*(X) 【cross product】_<BP>・ Z_<(p)> → H^*(X)_<(p)>coincides with the cycle map. One of the main results of Totaro's paper is that there is a space X = BG for which the kernel of cl contains p-torsion elements. Here the Chow ring of a classifying space BG is defined [Tol,To2] as the limit Lim_<m→∞>CH^*((c^m - s)/G) of a system of algebraic varieties where G acts on C^m - S freely and codim(S) → ∞ as m → ∞. The group Totaro uses is G = Z/2 × 2^<1+4>_+, where 2^<1+4>_+ is the extraspecial 2-group of order 32, which is isomorphic to the central product of two copies of the dihedral group D_8 of order 8. Similar facts hold for the extraspecial 2-groups G = 2^<1+2n>_+ of order 2<1+2n>.Totaro computed the Chow rings of abelian groups and symmetric groups in and he and Pandharipande determined the Chow rings of O(n), SO(2n+1) aud SO(4). For these cases the cycle maps c^~l are isomorphisms, namely, CH^*(BG)_<(2)> =^~ BP^*(BG) 【cross product】_BP・Z_<(2)>. Field also determined the Chow ring of BSO(2n), but its BP-theory is unknown for n > 3. Vezzosi has shown that d^~ is epimorphic for X = BPGL_3c, p = 3. Totaro also gives many interesting theorems to study CH*(BG) in

对于光滑复代数簇X，余维I代数圈的群CH^I（X）模有理等价组装成Chow环CH^*（X）= CH_iCH^I（X）。Totaro构造了一个映射c^^~ 1：CH^*（X）→ BP^*（X）_<BP>· Z_&lt;（p）&gt;，使得组成c 1：CH^I（X）_&lt;（p）&gt; → C^~ 1&gt; BP^*（X）_<BP>· Z_&lt;（p）&gt; → H^*（X）_&lt;（p）&gt;与循环映射重合.一个主要的结果，户太郎的文件是，有一个空间X = BG的核心cl包含p-扭转元素。定义分类空间BG的Chow环[Tol，To 2]为代数簇系的极限Lim_&lt;m→∞&gt;CH^*（（c^m-s）/G），其中G自由作用在C^m-S上，当m → ∞时codim（S）→ ∞. Totaro使用的群是G = Z/2 × 2^&lt;1+4&gt;_+，其中2^&lt;1+4&gt;_+是32阶的超特殊2-群，它同构于8阶二面体群D_8的两个副本的中心积。对于2&lt;1+2n&gt;阶的超特殊2-群G = 2^&lt;1 + 2n&gt;_+，Totaro在[1]中计算了交换群和对称群的Chow环，他和Pandharipande确定了O（n），SO（2n+1）和SO（4）的Chow环.对于这些情形，循环映射c^~l是同构的，即CH^*（BG）_&lt;（2）&gt; =^~ BP^*（BG）[叉积]_BP·Z_&lt;（2）&gt;. Field还确定了BSO（2n）的Chow环，但对于n &gt; 3，它的BP-理论是未知的。Vezzosi已经证明，对于X = BPGL_3c，p = 3，d^~是满态的。Totaro还给出了许多有趣的定理来研究CH*（BG），

项目成果

N.Yagita: "Chow ring of classifying spaces of extra p-groups"Contemp. Math.. 293. 397-403 (2002)

N.Yagita：“额外 p 群的分类空间的 Chow 环”Contemp。

Nyagita: "Chow ring of classifying spaces of extraspecial P-groups."Comtemp, Math.. 293. 397-403 (2002)

Nyagita：“超特殊 P 群的分类空间的 Chow 环。”Comtemp，Math.. 293. 397-403 (2002)

B.Schuster: "Transfers of Chern classes in BP-cohomology and Chow ring"Trans. AMS. 353. 1039-1054 (2001)

B.Schuster：“BP 上同调和 Chow 环中陈氏类的传递”Trans。

Darletaz, C.Ausoni, Mimura, N.Yagita: "Integral cohomology and chern classes of the special linear groups over the ring of integers."Math. Prcc. Cambridge Phil. Sec.. 131. 445-457 (2001)

Darletaz、C.Ausoni、Mimura、N.Yagita：“整数环上特殊线性群的积分上同调和陈类。”数学。

H.Andersen and M.Kaneda: "Filtrations on G*T-modules"Proc. London Math. Soc. 82. 614-646 (2001)

H.Andersen 和 M.Kaneda：“G*T 模块上的过滤”Proc。