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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

14C15 (Equivariant) Chow groups and rings; motives
14F30 \(p\)-adic cohomology, crystalline cohomology
14C05 Parametrization (Chow and Hilbert schemes)
Full Text: DOI
[1] Nagoya Math. J. 94 pp 1– (1984) · Zbl 0574.14005 · doi:10.1017/S002776300002081X
[2] Algebraic Geometry and Ring Theory pp 29– (1978)
[3] DOI: 10.1007/BF02684688 · Zbl 0221.14007 · doi:10.1007/BF02684688
[4] Lecture Notes in Mathematics 20 (1966)
[5] DOI: 10.2307/2007100 · Zbl 0068.34401 · doi:10.2307/2007100
[6] Homological Algebra (1956) · Zbl 0075.24305
[7] Ann. Sci. École Norm. Sup. 7 pp 181– (1974) · Zbl 0307.14008 · doi:10.24033/asens.1266
[8] Duke University Mathematics Series IV (1980)
[9] (1968)
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