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Relative cycles and Chow sheaves. (English) Zbl 1019.14004
Voevodsky, Vladimir et al., Cycles, transfers, and motivic homology theories. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 143, 10-86 (2000).
Let \(X@>f>>S\) be a scheme of finite type over a noetherian scheme \(S\). In this paper, the authors introduce a class of cycles on \(X\) which are called relative cycles on \(X\) over \(S\). Roughly speaking, a relative cycle on \(X\) over \(S\) is a cycle on \(X\) which lies over the generic points of \(S\) and has a well-defined specialization to any fibre of the morphism \(f\). If \(\text{Cycl} (X/S,r)\) denotes the group of relative cycles of relative dimension \(r\) on \(X\) over \(S\), then there is the subgroup \(z(X/S,r)\) of those relative cycles, whose specializations have integral coefficients. This construction defines a presheaf of abelian groups on the category of noetherian schemes over \(S\), which is also denoted by \(z(X/S,r)\) and called the \(r\)-th Chow presheaf. These Chow presheaves are the main objects of study in the course of the paper, and their theory is developed in full detail. The basic idea of this new theory is of topological origin (Lawson homology theory and singular homology of schemes) and based upon some earlier work by the authors themselves. At any rate, the theory of sheaves of relative cycles developed here is one of the main tools for constructing a motivic cohomology theory in the sense of A. Beilinson.
For the entire collection see [Zbl 1021.14006].

14C25 Algebraic cycles
14C15 (Equivariant) Chow groups and rings; motives