## Tag `02R5`

Chapter 41: Chow Homology and Chern Classes > Section 41.12: Proper pushforward

Lemma 41.12.2. Let $(S, \delta)$ be as in Situation 41.7.1. Let $X$, $Y$, and $Z$ be locally of finite type over $S$. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms. Then $g_* \circ f_* = (g \circ f)_*$ as maps $Z_k(X) \to Z_k(Z)$.

Proof.Let $W \subset X$ be an integral closed subscheme of dimension $k$. Consider $W' = f(Z) \subset Y$ and $W'' = g(f(Z)) \subset Z$. Since $f$, $g$ are proper we see that $W'$ (resp. $W''$) is an integral closed subscheme of $Y$ (resp. $Z$). We have to show that $g_*(f_*[W]) = (f \circ g)_*[W]$. If $\dim_\delta(W'') < k$, then both sides are zero. If $\dim_\delta(W'') = k$, then we see the induced morphisms $$ W \longrightarrow W' \longrightarrow W'' $$ both satisfy the hypotheses of Lemma 41.11.1. Hence $$ g_*(f_*[W]) = \deg(W/W')\deg(W'/W'')[W''], \quad (f \circ g)_*[W] = \deg(W/W'')[W'']. $$ Then we can apply Morphisms, Lemma 28.48.9 to conclude. $\square$

The code snippet corresponding to this tag is a part of the file `chow.tex` and is located in lines 1743–1749 (see updates for more information).

```
\begin{lemma}
\label{lemma-compose-pushforward}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$, $Y$, and $Z$ be locally of finite type over $S$.
Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms.
Then $g_* \circ f_* = (g \circ f)_*$ as maps $Z_k(X) \to Z_k(Z)$.
\end{lemma}
\begin{proof}
Let $W \subset X$ be an integral closed subscheme of dimension $k$.
Consider $W' = f(Z) \subset Y$ and $W'' = g(f(Z)) \subset Z$.
Since $f$, $g$ are proper we see that $W'$ (resp.\ $W''$) is
an integral closed subscheme of $Y$ (resp.\ $Z$).
We have to show that $g_*(f_*[W]) = (f \circ g)_*[W]$.
If $\dim_\delta(W'') < k$, then both sides are zero.
If $\dim_\delta(W'') = k$, then we see the induced morphisms
$$
W \longrightarrow
W' \longrightarrow
W''
$$
both satisfy the hypotheses of Lemma \ref{lemma-equal-dimension}. Hence
$$
g_*(f_*[W]) = \deg(W/W')\deg(W'/W'')[W''],
\quad
(f \circ g)_*[W] = \deg(W/W'')[W''].
$$
Then we can apply
Morphisms, Lemma \ref{morphisms-lemma-degree-composition}
to conclude.
\end{proof}
```

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