# The Stacks Project

## Tag 02R3

### 41.13. Proper pushforward

Definition 41.13.1. Let $(S, \delta)$ be as in Situation 41.8.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is proper.

1. Let $Z \subset X$ be an integral closed subscheme with $\dim_\delta(Z) = k$. We define $$f_*[Z] = \left\{ \begin{matrix} 0 & \text{if} & \dim_\delta(f(Z))< k, \\ \deg(Z/f(Z)) [f(Z)] & \text{if} & \dim_\delta(f(Z)) = k. \end{matrix} \right.$$ Here we think of $f(Z) \subset Y$ as an integral closed subscheme. The degree of $Z$ over $f(Z)$ is finite if $\dim_\delta(f(Z)) = \dim_\delta(Z)$ by Lemma 41.12.1.
2. Let $\alpha = \sum n_Z [Z]$ be a $k$-cycle on $X$. The pushforward of $\alpha$ as the sum $$f_* \alpha = \sum n_Z f_*[Z]$$ where each $f_*[Z]$ is defined as above. The sum is locally finite by Lemma 41.12.2 above.

By definition the proper pushforward of cycles $$f_* : Z_k(X) \longrightarrow Z_k(Y)$$ is a homomorphism of abelian groups. It turns $X \mapsto Z_k(X)$ into a covariant functor on the category of schemes locally of finite type over $S$ with morphisms equal to proper morphisms.

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. $\square$

Lemma 41.13.3. Let $(S, \delta)$ be as in Situation 41.8.1. Let $f : X \to Y$ be a proper morphism of schemes which are locally of finite type over $S$.

1. Let $Z \subset X$ be a closed subscheme with $\dim_\delta(Z) \leq k$. Then $$f_*[Z]_k = [f_*{\mathcal O}_Z]_k.$$
2. Let $\mathcal{F}$ be a coherent sheaf on $X$ such that $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. Then $$f_*[\mathcal{F}]_k = [f_*{\mathcal F}]_k.$$

Note that the statement makes sense since $f_*\mathcal{F}$ and $f_*\mathcal{O}_Z$ are coherent $\mathcal{O}_Y$-modules by Cohomology of Schemes, Proposition 29.19.1.

Proof. Part (1) follows from (2) and Lemma 41.11.3. Let $\mathcal{F}$ be a coherent sheaf on $X$. Assume that $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. By Cohomology of Schemes, Lemma 29.9.7 there exists a closed subscheme $i : Z \to X$ and a coherent $\mathcal{O}_Z$-module $\mathcal{G}$ such that $i_*\mathcal{G} \cong \mathcal{F}$ and such that the support of $\mathcal{F}$ is $Z$. Let $Z' \subset Y$ be the scheme theoretic image of $f|_Z : Z \to Y$. Consider the commutative diagram of schemes $$\xymatrix{ Z \ar[r]_i \ar[d]_{f|_Z} & X \ar[d]^f \\ Z' \ar[r]^{i'} & Y }$$ We have $f_*\mathcal{F} = f_*i_*\mathcal{G} = i'_*(f|_Z)_*\mathcal{G}$ by going around the diagram in two ways. Suppose we know the result holds for closed immersions and for $f|_Z$. Then we see that $$f_*[\mathcal{F}]_k = f_*i_*[\mathcal{G}]_k = (i')_*(f|_Z)_*[\mathcal{G}]_k = (i')_*[(f|_Z)_*\mathcal{G}]_k = [(i')_*(f|_Z)_*\mathcal{G}]_k = [f_*\mathcal{F}]_k$$ as desired. The case of a closed immersion is straightforward (omitted). Note that $f|_Z : Z \to Z'$ is a dominant morphism (see Morphisms, Lemma 28.6.3). Thus we have reduced to the case where $\dim_\delta(X) \leq k$ and $f : X \to Y$ is proper and dominant.

Assume $\dim_\delta(X) \leq k$ and $f : X \to Y$ is proper and dominant. Since $f$ is dominant, for every irreducible component $Z \subset Y$ with generic point $\eta$ there exists a point $\xi \in X$ such that $f(\xi) = \eta$. Hence $\delta(\eta) \leq \delta(\xi) \leq k$. Thus we see that in the expressions $$f_*[\mathcal{F}]_k = \sum n_Z[Z], \quad \text{and} \quad [f_*\mathcal{F}]_k = \sum m_Z[Z].$$ whenever $n_Z \not = 0$, or $m_Z \not = 0$ the integral closed subscheme $Z$ is actually an irreducible component of $Y$ of $\delta$-dimension $k$. Pick such an integral closed subscheme $Z \subset Y$ and denote $\eta$ its generic point. Note that for any $\xi \in X$ with $f(\xi) = \eta$ we have $\delta(\xi) \geq k$ and hence $\xi$ is a generic point of an irreducible component of $X$ of $\delta$-dimension $k$ as well (see Lemma 41.10.1). Since $f$ is quasi-compact and $X$ is locally Noetherian, there can be only finitely many of these and hence $f^{-1}(\{\eta\})$ is finite. By Morphisms, Lemma 28.48.1 there exists an open neighbourhood $\eta \in V \subset Y$ such that $f^{-1}(V) \to V$ is finite. Replacing $Y$ by $V$ and $X$ by $f^{-1}(V)$ we reduce to the case where $Y$ is affine, and $f$ is finite.

Write $Y = \mathop{\rm Spec}(R)$ and $X = \mathop{\rm Spec}(A)$ (possible as a finite morphism is affine). Then $R$ and $A$ are Noetherian rings and $A$ is finite over $R$. Moreover $\mathcal{F} = \widetilde{M}$ for some finite $A$-module $M$. Note that $f_*\mathcal{F}$ corresponds to $M$ viewed as an $R$-module. Let $\mathfrak p \subset R$ be the minimal prime corresponding to $\eta \in Y$. The coefficient of $Z$ in $[f_*\mathcal{F}]_k$ is clearly $\text{length}_{R_{\mathfrak p}}(M_{\mathfrak p})$. Let $\mathfrak q_i$, $i = 1, \ldots, t$ be the primes of $A$ lying over $\mathfrak p$. Then $A_{\mathfrak p} = \prod A_{\mathfrak q_i}$ since $A_{\mathfrak p}$ is an Artinian ring being finite over the dimension zero local Noetherian ring $R_{\mathfrak p}$. Clearly the coefficient of $Z$ in $f_*[\mathcal{F}]_k$ is $$\sum\nolimits_{i = 1, \ldots, t} [\kappa(\mathfrak q_i) : \kappa(\mathfrak p)] \text{length}_{A_{\mathfrak q_i}}(M_{\mathfrak q_i})$$ Hence the desired equality follows from Algebra, Lemma 10.51.12. $\square$

The code snippet corresponding to this tag is a part of the file chow.tex and is located in lines 3463–3663 (see updates for more information).

\section{Proper pushforward}
\label{section-proper-pushforward}

\begin{definition}
\label{definition-proper-pushforward}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$, $Y$ be locally of finite type over $S$.
Let $f : X \to Y$ be a morphism.
Assume $f$ is proper.
\begin{enumerate}
\item Let $Z \subset X$ be an integral closed subscheme
with $\dim_\delta(Z) = k$. We define
$$f_*[Z] = \left\{ \begin{matrix} 0 & \text{if} & \dim_\delta(f(Z))< k, \\ \deg(Z/f(Z)) [f(Z)] & \text{if} & \dim_\delta(f(Z)) = k. \end{matrix} \right.$$
Here we think of $f(Z) \subset Y$ as an integral closed subscheme.
The degree of $Z$ over $f(Z)$ is finite if
$\dim_\delta(f(Z)) = \dim_\delta(Z)$
by Lemma \ref{lemma-equal-dimension}.
\item Let $\alpha = \sum n_Z [Z]$ be a $k$-cycle on $X$. The
{\it pushforward} of $\alpha$ as the sum
$$f_* \alpha = \sum n_Z f_*[Z]$$
where each $f_*[Z]$ is defined as above. The sum is locally finite
by Lemma \ref{lemma-quasi-compact-locally-finite} above.
\end{enumerate}
\end{definition}

\noindent
By definition the proper pushforward of cycles
$$f_* : Z_k(X) \longrightarrow Z_k(Y)$$
is a homomorphism of abelian groups. It turns $X \mapsto Z_k(X)$
into a covariant functor on the category of schemes locally of
finite type over $S$ with morphisms equal to proper morphisms.

\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}

\begin{lemma}
\label{lemma-cycle-push-sheaf}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $f : X \to Y$ be a proper morphism of schemes which are
locally of finite type over $S$.
\begin{enumerate}
\item Let $Z \subset X$ be a closed subscheme with $\dim_\delta(Z) \leq k$.
Then
$$f_*[Z]_k = [f_*{\mathcal O}_Z]_k.$$
\item Let $\mathcal{F}$ be a coherent sheaf on $X$ such that
$\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. Then
$$f_*[\mathcal{F}]_k = [f_*{\mathcal F}]_k.$$
\end{enumerate}
Note that the statement makes sense since $f_*\mathcal{F}$ and
$f_*\mathcal{O}_Z$ are coherent $\mathcal{O}_Y$-modules by
Cohomology of Schemes, Proposition
\ref{coherent-proposition-proper-pushforward-coherent}.
\end{lemma}

\begin{proof}
Part (1) follows from (2) and Lemma \ref{lemma-cycle-closed-coherent}.
Let $\mathcal{F}$ be a coherent sheaf on $X$.
Assume that $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$.
By Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-support-closed}
there exists a closed subscheme $i : Z \to X$ and a coherent
$\mathcal{O}_Z$-module $\mathcal{G}$ such that
$i_*\mathcal{G} \cong \mathcal{F}$ and such that the support
of $\mathcal{F}$ is $Z$. Let $Z' \subset Y$ be the scheme theoretic image
of $f|_Z : Z \to Y$. Consider the commutative diagram of schemes
$$\xymatrix{ Z \ar[r]_i \ar[d]_{f|_Z} & X \ar[d]^f \\ Z' \ar[r]^{i'} & Y }$$
We have $f_*\mathcal{F} = f_*i_*\mathcal{G} = i'_*(f|_Z)_*\mathcal{G}$
by going around the diagram in two ways. Suppose we know the result holds
for closed immersions and for $f|_Z$. Then we see that
$$f_*[\mathcal{F}]_k = f_*i_*[\mathcal{G}]_k = (i')_*(f|_Z)_*[\mathcal{G}]_k = (i')_*[(f|_Z)_*\mathcal{G}]_k = [(i')_*(f|_Z)_*\mathcal{G}]_k = [f_*\mathcal{F}]_k$$
as desired. The case of a closed immersion is straightforward (omitted).
Note that $f|_Z : Z \to Z'$ is a dominant morphism (see
Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}).
Thus we have reduced to the case where
$\dim_\delta(X) \leq k$ and $f : X \to Y$ is proper and dominant.

\medskip\noindent
Assume $\dim_\delta(X) \leq k$ and $f : X \to Y$ is proper and dominant.
Since $f$ is dominant, for every irreducible component $Z \subset Y$
with generic point $\eta$ there exists a point $\xi \in X$ such
that $f(\xi) = \eta$. Hence $\delta(\eta) \leq \delta(\xi) \leq k$.
Thus we see that in the expressions
$$f_*[\mathcal{F}]_k = \sum n_Z[Z], \quad \text{and} \quad [f_*\mathcal{F}]_k = \sum m_Z[Z].$$
whenever $n_Z \not = 0$, or $m_Z \not = 0$ the integral closed
subscheme $Z$ is actually an irreducible component of $Y$ of
$\delta$-dimension $k$. Pick such an integral closed subscheme
$Z \subset Y$ and denote $\eta$ its generic point. Note that for
any $\xi \in X$ with $f(\xi) = \eta$ we have $\delta(\xi) \geq k$
and hence $\xi$ is a generic point of an irreducible component
of $X$ of $\delta$-dimension $k$ as well
(see Lemma \ref{lemma-multiplicity-finite}). Since $f$ is quasi-compact
and $X$ is locally Noetherian, there can be only finitely many of
these and hence $f^{-1}(\{\eta\})$ is finite.
By Morphisms, Lemma \ref{morphisms-lemma-generically-finite} there exists
an open neighbourhood $\eta \in V \subset Y$ such that $f^{-1}(V) \to V$
is finite. Replacing $Y$ by $V$ and $X$ by $f^{-1}(V)$ we reduce to the
case where $Y$ is affine, and $f$ is finite.

\medskip\noindent
Write $Y = \Spec(R)$ and $X = \Spec(A)$ (possible as
a finite morphism is affine).
Then $R$ and $A$ are Noetherian rings and $A$ is finite over $R$.
Moreover $\mathcal{F} = \widetilde{M}$ for some finite $A$-module
$M$. Note that $f_*\mathcal{F}$ corresponds to $M$ viewed as an $R$-module.
Let $\mathfrak p \subset R$ be the minimal prime corresponding
to $\eta \in Y$. The coefficient of $Z$ in $[f_*\mathcal{F}]_k$
is clearly $\text{length}_{R_{\mathfrak p}}(M_{\mathfrak p})$.
Let $\mathfrak q_i$, $i = 1, \ldots, t$ be the primes of $A$
lying over $\mathfrak p$. Then $A_{\mathfrak p} = \prod A_{\mathfrak q_i}$
since $A_{\mathfrak p}$ is an Artinian ring being finite over the
dimension zero local Noetherian ring $R_{\mathfrak p}$.
Clearly the coefficient of $Z$ in $f_*[\mathcal{F}]_k$ is
$$\sum\nolimits_{i = 1, \ldots, t} [\kappa(\mathfrak q_i) : \kappa(\mathfrak p)] \text{length}_{A_{\mathfrak q_i}}(M_{\mathfrak q_i})$$
Hence the desired equality follows from
Algebra, Lemma \ref{algebra-lemma-pushdown-module}.
\end{proof}

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