| Co-Investigator(Kenkyū-buntansha) |
OKAYASU Takashi Ibaraki Univ., the College of Education, associate Professor, 教育学部, 助教授 (00191958)
KUDOU Kenzi Ibaraki Univ., the College of Education, Lecture, 教育学部, 講師 (00114017)
KANEDA Masaharu Osaka City Univ., Faculty of Nat.Science, Professor, 理学部, 教授 (60204575)
TEZUKA Mithishige Ryuukyuu Univ., Faculty of Nat.Science, Professor, 理学部, 教授 (20197784)
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| Research Abstract |
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.
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