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

第一个结果是引入一个新的同调理论Voevodsky的categoryDM（k）的动机在一个域k，它推广了Gillet-Soul 'e的重量同调，和Kato-Suslin重量同调的Geisser。作为Bondarko定理的一个结果，我们可以得到以下范畴的等价性。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，Chow Morives范畴，由Chow（k）到DM（k）的包含导出的一个等价性，当限制到满足条件H（M（X）[n]）= 0时，X是光滑的和投射的，并且n> 0。此外，这个函子恢复和推广了Gillet-Soul 'e的权同调和Geisser的Kato-Suslin权同调。第二个结果是比较motivic同调和' etale motivic同调，特别是在有限域上。有一个从动机M的动机同调到它的标准动机同调的典范态射：H ^M_i（C）\\to H ^{M，et}_i（C）.定理：设p = ch（k）.如果k是代数闭的，则\\alpha ^* 是逆p后的同构。如果k是有限的，则可以根据Gillet-Soul'e模p-挠的权同调来计算\\alpha的核和上核。

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.