## 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].

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0H6T. Beware of the difference between the letter 'O' and the digit '0'.