正標数の体上のモチフィックコホモロジーのホモトピー論的研究

第一个结果是对域k上动机的Voevodsky范畴dm (k)引入了一个新的同调理论，推广了Gillet-Soul'e的权同调和Geisser的Kato-Suslin权同调。作为Bondarko定理的一个结果，我们可以得到如下的范畴等价。定理：从DM(k)上的同调函子范畴到Chow(k)上的可加函子范畴的函子，即由Chow(k)包含到DM(k)中导出的Chow动机范畴，当约束于同调函子的满子范畴H时，在满足H(M(X)[n]) = 0的条件下，当X是光滑射影且n>时，可导出等价。此外，该函子恢复并推广了Gillet-Soul'e的权同源性和Geisser的Kato-Suslin权同源性。第二个结果是比较动机同调与“假动机同调”，特别是在有限域上。从一个动机M的动机同构到它的另一个动机同构的正则态射：H^M_i(C)到H^{M， et}_i(C)。定理：设p=ch(k)如果k是代数封闭的，则对p求逆后的\\ α ^*是同构的。如果k是有限的，则可以根据Gillet-Soul'e模p-扭转的权同调来计算\\ α的核和核。

The first result is to introduce a new homology theory for Voevodsky's categoryDM(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.