The Stacks project

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:

1. 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.

2. 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.

3. 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 $c_{equi}(X/S, r)$.

4. Base change, flat pullback, and proper pushforward as defined in [SV] agrees with ours when both apply.

5. 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.

6. 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 opration $Cor_{X/Y}(-, -)$ in [Corollary 3.7.5, SV].