generalized cohomology and classifying spases

设p是一个素数，k是复数域C的子域，其中包含一个原始的1的p次方根。对于k上有限型格式X，用Suslin-Voevodsky （[Su-Vo],[Vo2]）构造了动机上同调H^<*,*> (X; Z/p) =【对称】_<m,n>H^<*,*> (X; Z/p)。对于光滑X，上同调H^<2*,*> (X; Z/p) =【对称】_nH^<2n,n> (X; Z/p)被X上代数环的经典模p Chow环CH^* (X)/p所识别。包含t_C: k∧C推导出一个自然变换（实现映射）t^<m,n>_C: H^<m,n>→H^m(X(C)；Z/p)，其中X(C)是C值点的复变。让我们写出共像（1.1）h^<*,*>(X; Z/p) =【对称】_<m,n> h^< m,n>(X; Z/p)/Ker（t^<m,n>_C）已知存在一个元素r∈h^< 0,1>(Speck(k)；Z/p) t^*_C(r) = 1。然后我们得到了广义代数单态（1.2）h^<*,*>(XZ/p)〓h^ * (X(C)；Z/p)【叉积】Z/p[r，r^<-1>]其中x∈H^n (x (C)；Z/p)由（n,n）给出。当k C和Beilinson-Lichtenbaum猜想对p成立时，我们也有注入II^*(X; Z/p)【外积】Z/p[r]〓h^<*,*>(X; Z/p)。其中t^<m,n>_C: H^<m,n>(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，且B(m,p)-条件成立。那么这个重阶代数h^<*,*> (X; Z/p)与gr h^ *(X(C)是同构的双度模；Z/p)【叉积】Z/p[r]通过过滤F_1 = Im（t^<*,i>_C）且0≠r∈H^<0,1>(Spec(C)；Z/p) = Z/p，而乘法是不同的。

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.