Lemma 62.14.1. If $\alpha $ and $\beta $ are relative cycles, then so is $\alpha \circ \beta $.

## 62.14 Composition of relative cycles

Let $S$ be a locally Noetherian scheme. Let $X \to Y$ be a morphism of schemes locally of finite type over $S$. We are going to define a map

using the construction in Section 62.13. We already know the construction is bilinear (Lemma 62.13.1) hence we obtain the displayed arrow once we show the following.

**Proof.**
The formation of $\alpha \circ \beta $ is compatible with base change by Lemma 62.13.3. Thus we may assume $S$ is the spectrum of a discrete valuation ring with generic point $\eta $ and closed point $0$ and we have to show that $sp_{X/S}((\alpha \circ \beta )_\eta ) = (\alpha \circ \beta )_0$. Since we are trying to prove an equality of cycles, we may work locally on $Y$ and $X$ (this uses Lemmas 62.13.2 and 62.4.4 to see that the constructions commute with restriction). Thus we may assume $X$ and $Y$ are affine. By Lemma 62.6.9 we can find a completely decomposed proper morphism $g : Y' \to Y$ such that $g^*\alpha $ is in the image of (62.6.8.1).

Since the family of morphisms $g_\eta : Y'_\eta \to Y_\eta $ is completely decomposed, we can find $\beta '_\eta \in Z_ e(Y'_\eta )$ such that $\beta _\eta = \sum g_{\eta , *}\beta '_\eta $, see Chow Homology, Lemma 42.22.4. Set $\beta '_0 = sp_{Y'/S}(\beta '_\eta )$ so that $\beta ' = (\beta '_\eta , \beta '_0)$ is a relative $e$-cycle on $Y'/S$. Then $g_*\beta '$ and $\beta $ are relative $e$-cycles on $Y/S$ (Lemma 62.6.2) which have the same value at $\eta $ and hence are equal (Lemma 62.6.6). By linearity (Lemma 62.13.1) it suffices to show that $\alpha \circ g_*\beta '$ is a relative $(r + e)$-cycle.

Set $X' = X \times _ Y Y'$ and denote $f : X' \to X$ the projection. By Lemma 62.13.6 we see that $\alpha \circ g_*\beta ' = f_*(g^*\alpha \circ \beta ')$. By Lemma 62.6.2 it suffices to show that $g^*\alpha \circ \beta '$ is a relative $(r + e)$-cycle. Using Lemma 62.6.10 and bilinearity this reduces us to the case discussed in the next paragraph.

Assume $\alpha = [Z/X/Y]_ r$ and $\beta = [W/Y/S]$ where $Z \subset X$ is a closed subscheme flat and of relative dimension $\leq r$ over $Y$ and $W \subset Y$ is a closed subscheme flat and of relative dimension $\leq e$ over $S$. By Lemma 62.13.5 we see that

and $Z \times _ X W \subset X$ is a closed subscheme flat over $S$ of relative dimension $\leq r + e$. This is a relative $(r + e)$-cycle by Lemma 62.6.8. $\square$

Lemma 62.14.2. Let $f : X \to Y$ and $g : Y \to S$ be a morphisms of schemes. Assume $S$ locally Noetherian, $g$ locally of finite type and flat of relative dimension $e \ge 0$, and $f$ locally of finite type and flat of relative dimension $r \geq 0$. Then $[X/X/Y]_ r \circ [Y/Y/S]_ e = [X/X/S]_{r + e}$ in $z(X/S, r + e)$.

**Proof.**
Special case of Lemma 62.13.5.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)