2 Spencer Bloch And
نویسنده
چکیده
Let X be a smooth, projective algebraic variety over an algebraically closed field k, and let H(X) denote a Weil cohomology theory. The existence of algebraic cycles on X×X inducing as correspondences the various Künneth projectors π : H(X) → H (X) is one of the standard conjectures of Grothendieck, [8], [9]. It is known in general only for the cases i = 0, 1, 2d − 1, 2d where d = dimX. The purpose of this note is to consider correspondences on smooth quasi-projective varieties U . In the first section we prove the existence of an “algebraic” Künneth projector π : H(U) → H(U) assuming that U admits a smooth, projective completion X. The word algebraic is placed in quotes here because in fact the algebraic cycle on X × U inducing π is not, as one might imagine, trivialized on (X − U) × U . It is only partially trivialized. This partial trivialization is sufficient to define a class in H c (U)⊗H (U) giving the desired projection. Of course, our cycle on X × U will be trivialized on (X − U) × V for V ⊂ U suitably small nonempty open, but our method does not in any obvious way yield a full trivialization on (X − U) × U . We finish this first section with some comments on π for i > 1 and some speculation, mostly coming from discussions with A. Beilinson, on how these ideas might be applied to study the Milnor conjecture that the Galois cohomology ring of the function field H(k(X),Z/nZ) is generated by H. In the last section, we use the existence of relative motivic cohomology [10] to prove an integrality and independence of l result for the trace of an
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تاریخ انتشار 2005