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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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