Abstract

In this paper we study in detail the so-called Chow-weight homology of Voevodsky motivic complexes and relate it to motivic homology. We generalize earlier results and prove that the vanishing of higher motivic homology groups of a motif M implies similar vanishing for its Chow-weight homology along with effectivity properties of the higher terms of its weight complex t(M) and of higher Deligne weight quotients of its cohomology. Applying this statement to motives with compact support we obtain a similar relation between the vanishing of Chow groups and the cohomology with compact support of varieties. Moreover, we prove that if higher motivic homology groups of a geometric motif or a variety over a universal domain are torsion (in a certain “range”) then the exponents of these groups are uniformly bounded. To prove our main results we study Voevodsky slices of motives. Since the slice functors do not respect the compactness of motives, the results of the previous Chow-weight homology paper are not sufficient for our purposes; this is our main reason to extend them to (\({{w}_{{{\text{Chow}}}}}\)-bounded below) motivic complexes.

中文翻译：

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动机复合体的周重同调性及其与同调性的关系

摘要

在本文中，我们详细研究了Voevodsky动机复合体的所谓Chow-weight同源性，并将其与动机同源性联系起来。我们归纳了较早的结果并证明，图案M的较高动机同质性基团的消失暗示其Chow-weight同源性也具有相似的消失性，以及其权重复杂度t（M）以及其同调性较高的Deligne重量商。将这一陈述应用于具有紧密支持的动机，我们在松散的周氏族群与具有紧密支持的品种的同调之间获得了相似的关系。此外，我们证明，如果几何图案或通用域上的多种高动机同质基团是扭转的（在某个“范围内”），则这些基团的指数是一致有界的。为了证明我们的主要结果，我们研究了Voevodsky动机的各个方面。由于切片函子不尊重动机的紧凑性，因此以前的Chow-weight同源性论文的结果不足以满足我们的目的。这是我们将其扩展到（\（{{w} _ {{\ text {Chow}}}}} \\）-边界以下的动机复合体的主要原因。