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Tag 02R5

Chapter 41: Chow Homology and Chern Classes > Section 41.13: Proper pushforward

Lemma 41.13.2. Let $(S, \delta)$ be as in Situation 41.8.1. 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 41.12.1. 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 3507–3513 (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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