generalized cohomology and classifying spases

设p是素数，k是复数域C的子域，复数域C包含1的本原p次根。对于k上有限类型的方案X，由Suslin-Voevodsky([SU-Vo]，[Vo2])构造了模上同调H^&lt；*，*&gt；(X；Z/p)=[对称性]_&lt；m，n&gt；H^&lt；*，*&gt；(X；Z/p)。对于光滑X，上同调H^&lt；2*，*&gt；(X；Z/p)=[对称性]_nH^&lt；2n，n&gt；(X；Z/p)与X上经典的代数圈的mod p-Chow环CH^*(X)/p一致。包含t_C：K⊂C诱导一个自然变换(实现映射)t^&lt；m，n&gt；_C：H^&lt；m，n&gt；→H^m(X(C)；Z/p)，其中X(C)是C值点的复变。设(1.1)h^&lt；*，*&gt；(X；Z/p)=[对称性]_&lt；m，n&gt；H^&lt；m，n&gt；(X；Z/p)/Ker(t^&lt；m，n&gt；_C)已知存在元素r∈H^&lt；0，1&gt；(SPECK(K)；Z/p)，t^*_C(R)=1，则(1.2)h^&lt；*，*&gt；(Xz/p)〓H^*(X(C)；Z/p)[叉积]Z/p[r，r^&lt；-1&gt；]其中x∈H^n(X(C)；Z/p)的二次由(n，n)给出。当k C和Beilinson-Lichtenbaum猜想对p成立时，我们还有内射II^*(X；Z/p)[叉积]Z/p[r]〓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)是到X的C值点X(C)的实现映射。假设k=C且B(m，p)-条件成立。然后这个二次代数h^&lt；*，*&gt；(X；Z/p)通过滤子F_1=Im(t^&lt；*，i&gt；_C)和0≠r∈H^&lt；0，1&gt；(Spec(C)；Z/p)〓Z/p同构于GR H^*(X(C)；Z/p)[叉积]Z/p，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.