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A note on cubic equivalences. (English) Zbl 0594.14008
Let T be a (complex) smooth quasi-projective variety, V a smooth projective variety of dimension m, p an integer $$\geq 0$$, $$r=m-p$$ and Z a cycle of codimension p in $$T\times V$$. We have natural maps $$\{^ tZ\}:\quad gr^ rH^{2r+\ell}(V,{\mathbb{Z}})\to gr^ 0H^{\ell '}(T,{\mathbb{Z}}),$$ where $$t\in T$$, $$\ell '\geq 0$$ is an integer and gr is associated to the filtration given by codimension (coniveau filtration). The main result is: if for a fixed integer $$\ell \geq 0$$ all the cycles Z(t) are $$\ell$$-cube equivalent to zero, then the above maps are zero for $$\ell '<\ell$$. Similar statements are given in the abstract case, working with (étale) de Rham cohomology (in characteristic 0) or with $$\nu$$- adic cohomology (in characteristic $$\neq 0)$$. For the definition of cubic equivalence relation, as well as for the similar result over $${\mathbb{C}}$$ concerning Hodge cohomology, see the author, Nagoya Math. J. 94, 1-41 (1984; Zbl 0574.14005).
Reviewer: C.Bănică

##### MSC:
 14C15 (Equivariant) Chow groups and rings; motives 14F30 $$p$$-adic cohomology, crystalline cohomology 14C05 Parametrization (Chow and Hilbert schemes)
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##### References:
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