| 研究実績の概要 |
The first result is to introduce a new homology theory for Voevodsky's category
DM(k) of motives over a field k, which generalises the weight homology of Gillet-Soul'e, and the Kato-Suslin weight homology of Geisser. As a consequence of a theorem of Bondarko, we can obtain the following equivalence of categories.
Theorem: The functor from the category of homological functors on DM(k) to the category of additive functors on Chow(k), the category of Chow morives, which is induced by the inclusion of Chow(k) to DM(k) induces an equivalence when restricted to the full subcategory of homological functors H satisfying the condition that H(M(X)[n]) = 0 when X is smooth and projective and n>0. Moreover, this functor recovers and generalises Gillet-Soul'e's weight homology, and the Kato-Suslin weight homology of Geisser.
The second result is to compare motivic homology with 'etale motivic homology, in particular over a finite field. There is a canonical morphism from motivic homology of a motive M to its 'etale motivic homology \\alpha: H^M_i(C) \\to H^{M, et}_i(C).
Theorem: Let p=ch(k). If k is algebraically closed the \\alpha^* is an isomorphism after inverting p. If k is finite, then one can compute the kernel and cokernel of \\alpha in terms of the weight homology of Gillet-Soul'e modulo p-torsion.
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