The Stacks project

81.8 Proper pushforward

This section is the analogue of Chow Homology, Section 42.12.

Definition 81.8.1. In Situation 81.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is proper.

  1. Let $Z \subset X$ be an integral closed subspace with $\dim _\delta (Z) = k$. Let $Z' \subset Y$ be the image of $Z$ as in Lemma 81.7.1. We define

    \[ f_*[Z] = \left\{ \begin{matrix} 0 & \text{if} & \dim _\delta (Z')< k, \\ \deg (Z/Z') [Z'] & \text{if} & \dim _\delta (Z') = k. \end{matrix} \right. \]

    The degree of $Z$ over $Z'$ is defined and finite if $\dim _\delta (Z') = \dim _\delta (Z)$ by Lemma 81.7.4 and Spaces over Fields, Definition 71.5.2.

  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 81.7.5 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 whose object are good algebraic spaces over $B$ and whose morphisms are proper morphisms over $B$.

Lemma 81.8.2. In Situation 81.2.1 let $X, Y, Z/B$ be good. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms over $B$. 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 subspace of dimension $k$. Consider the integral closed subspaces $W' \subset Y$ and $W'' \subset Z$ we get by applying Lemma 81.7.1 to $f$ and $W$ and then to $g$ and $W'$. Then $W \to W'$ and $W' \to W''$ are surjective and proper. 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 $W \to W'$ and $W' \to W''$ both satisfy the hypotheses of Lemma 81.7.4. Hence

\[ g_*(f_*[W]) = \deg (W/W')\deg (W'/W'')[W''], \quad (f \circ g)_*[W] = \deg (W/W'')[W'']. \]

Then we can apply Spaces over Fields, Lemma 71.5.3 to conclude. $\square$

Lemma 81.8.3. In Situation 81.2.1 let $f : X \to Y$ be a proper morphism of good algebraic spaces over $B$.

  1. Let $Z \subset X$ be a closed subspace 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 Spaces, Lemma 68.20.2.

Proof. Part (1) follows from (2) and Lemma 81.6.3. Let $\mathcal{F}$ be a coherent sheaf on $X$. Assume that $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k$. By Cohomology of Spaces, Lemma 68.12.7 there exists a closed immersion $i : Z \to X$ 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 $f|_ Z : Z \to Y$, see Morphisms of Spaces, Definition 66.16.2. Consider the commutative diagram

\[ \xymatrix{ Z \ar[r]_ i \ar[d]_{f|_ Z} & X \ar[d]^ f \\ Z' \ar[r]^{i'} & Y } \]

of algebraic spaces over $B$. Observe that $f|_ Z$ is surjective (follows from Morphisms of Spaces, Lemma 66.16.3 and the fact that $|f|$ is closed) and proper (follows from Morphisms of Spaces, Lemmas 66.40.3, 66.40.5, and 66.40.6). 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 follows from Lemma 81.4.3 and the definitions. Thus we have reduced to the case where $\dim _\delta (X) \leq k$ and $f : X \to Y$ is proper and surjective.

Assume $\dim _\delta (X) \leq k$ and $f : X \to Y$ is proper and surjective. 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 subspace $Z$ is actually an irreducible component of $Y$ of $\delta $-dimension $k$ (see Lemma 81.4.5). Pick such an integral closed subspace $Z \subset Y$ 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 81.4.5). By Spaces over Fields, Lemma 71.3.2 there exists an open subspace $\eta \in V \subset Y$ such that $f^{-1}(V) \to V$ is finite. Since $\eta $ is a generic point of an irreducible component of $|Y|$ we may assume $V$ is an affine scheme, see Properties of Spaces, Proposition 65.13.3. Replacing $Y$ by $V$ and $X$ by $f^{-1}(V)$ we reduce to the case where $Y$ is affine, and $f$ is finite. In particular $X$ and $Y$ are schemes and we reduce to the corresponding result for schemes, see Chow Homology, Lemma 42.12.4 (applied with $S = Y$). $\square$

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