generalized cohomology and classifying spases

设p是素数，k是复数域C的子域，其中包含1的本原p次根。对k上有限型方案X，Suslin-Voevodsky（[Su-Vo]，[Vo2]）构造了运动上同调H ^<*，*>（X; Z/p）=[symmetry]_<m，n> H ^<*，*>（X; Z/p）.对光滑的X，上同调H ^<2 *，*>（X; Z/p）=[对称性]_nH ^<2n，n>（X; Z/p）与X上代数圈的经典模p Chow环CH ^*（X）/p相同.包含t_C：k <$C诱导了一个自然变换（实现映射）t ^<m，n>_C：H ^<m，n> → H ^m（X（C）; Z/p）其中X（C）是C值点的复簇。记为（1.1）h ^<*，*>（X; Z/p）=[symmetry]_<m，n> H ^<m，n>（X; Z/p）/Ker（t ^<m，n>_C）已知存在元素r ∈ H ^<0，1>（Speck（k）; Z/p），其中t ^*_C（r）= 1。然后我们有双阶代数单态（1.2）h ^&lt;*，*&gt;（XZ/p）&lt;$H ^*（X（C）; Z/p）[叉积] Z/p [r，r ^]其中x ∈ H ^n（X（C）; Z/p）的双阶由（n，n）给出。<-1>当k C和Beilinson-Lichtenbaum猜想对p成立时，我们也有注入II ^*（X; Z/p）[叉积] Z/p [r]&lt;$h ^&lt;*，*&gt;（X; Z/p）.其中t ^&lt;m，n&gt;_C：H ^&lt;m，n&gt;（X; Z/p）→ H ^m（X（C）; Z/p）是X到C值点X（C）的实现映射。假设k = C，并且B（m，p）条件成立。则这个双阶代数h ^&lt;*，*&gt;（X; Z/p）通过滤子F_1 = Im（t ^&lt;*，i&gt;_C）和0 &lt;$r ∈ H ^&lt;0，1&gt;（Spec（C）; Z/p）&lt;$Z/p同构为gr H ^*（X（C）; Z/p）[叉积] Z/p [r]的双度模，但乘法不同.

Let p be a prime number and k a subfield of the complex number field C, which contains a primitive p-th root of 1. For a scheme X of finite type over k, the motivic cohomology H^<*,*> (X ; Z/p) = 【symmetry】_<m,n>H^<*,*> (X ; Z/p) is constructed by Suslin-Voevodsky ([Su-Vo],[Vo2]). For a smooth X, the cohomology H^<2*,*> (X ; Z/p) = 【symmetry】_nH^<2n,n> (X ; Z/p) is identified with CH^* (X)/p the classical mod p Chow ring of algebraic cyles on X.The inclusion t_C : k ⊂ C induces a natural transformation (realization map) t^<m,n>_C : H^<m,n> → H^m(X(C) ; Z/p) where X(C) is the complex variety of C-valued points. Let us write the coimage(1.1)h^<*,*>(X ; Z/p) = 【symmetry】_<m,n>H^<m,n>(X ; Z/p)/Ker(t^<m,n>_C)It is known that there is an element r ∈ H^<0,1>(Speck(k) ; Z/p) with t^*_C(r) = 1. Then we have the bigraded algebra monomorphism(1.2)h^<*,*>(XZ/p) 〓 H^* (X(C) ; Z/p)【cross product】Z/p[r,r^<-1>]where the bidegree of x ∈ H^n (X(C) ; Z/p) is given by (n,n). When k C and the Beilinson-Lichtenbaum conjecture is true for p, we also have the injection II^*(X ; Z/p)【cross product】Z/p[r] 〓 h^<*,*>(X ; Z/p).where t^<m,n>_C : H^<m,n>(X ; Z/p) → H^m(X(C) ; Z/p) is the realization map to C-valued points X(C) of X. Suppose that k = C and the B(m,p)-condition holds. Then this bigraded algebra h^<*,*> (X ; Z/p) is isomorphic as bidegree modules to gr H^*(X(C) ; Z/p)【cross product】Z/p[r] by the filtration F_1 = Im(t^<*,i>_C) and 0 ≠ r ∈ H^<0,1>(Spec(C) ; Z/p) 〓 Z/p, while the multiplications are different.