62.15 Comparison with Suslin and Voevodsky
We have tried to use the same notation as in [SV], except that our notation for cycles is taken from Chow Homology, Section 42.8 ff. Here is a comparison:
In [Section 3.1, SV] there is a notion of a “relative cycle”, of a “relative cycle of dimension r”, and of a “equidimensional relative cycle of dimension r”. There is no corresponding notion in this chapter. Consequently, the groups Cycl(X/S, r), Cycl_{equi}(X/S, r), PropCycl(X/S, r), and PropCycl_{equi}(X/S, r), have no counter parts in this chapter.
On the bottom of [page 36, SV] the groups z(X/S, r), c(X/S, r), z_{equi}(X/S, r), c_{equi}(X/S, r) are defined. These agree with our notions when S is separated Noetherian and X \to S is separated and of finite type.
In [SV] the symbol z(X/S, r) is sometimes used for the presheaf S' \mapsto z(S' \times _ S X/S', r) on the category of schemes of finite type over S. Similarly for c(X/S, r), z_{equi}(X/S, r), and c_{equi}(X/S, r).
Base change, flat pullback, and proper pushforward as defined in [SV] agrees with ours when both apply.
For \alpha \in z(X/S, r) the operation \alpha \cap - : Z_ e(S) \to Z_{e + r}(X) defined in Section 62.11 agrees with the operation Cor(\alpha , -) in [Section 3.7, SV] when both are defined.
For X \to Y \to S the composition law z(X/Y, r) \otimes _\mathbf {Z} z(Y/S, e) \longrightarrow z(X/S, r + e) defined in Section 62.14 agrees with the operation Cor_{X/Y}(-, -) in [Corollary 3.7.5, SV].
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