The Stacks project

41.12 Proper pushforward

Definition 41.12.1. Let $(S, \delta )$ be as in Situation 41.7.1. 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 41.11.1.

  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 41.11.2 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 41.12.2. 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 28.49.9 to conclude. $\square$

A closed immersion is proper. If $i : Z \to X$ is a closed immersion then the maps

\[ i_* : Z_ k(Z) \longrightarrow Z_ k(X) \]

are all injective.

Lemma 41.12.3. Let $(S, \delta )$ be as in Situation 41.7.1. Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \subset X$ be closed subschemes such that $X = X_1 \cup X_2$ set theoretically. For every $k \in \mathbf{Z}$ the sequence of abelian groups

\[ \xymatrix{ Z_ k(X_1 \cap X_2) \ar[r] & Z_ k(X_1) \oplus Z_ k(X_2) \ar[r] & Z_ k(X) \ar[r] & 0 } \]

is exact. Here $X_1 \cap X_2$ is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by $-1$.

Proof. First assume $X$ is quasi-compact. Then $Z_ k(X)$ is a free $\mathbf{Z}$-module with basis given by the elements $[Z]$ where $Z \subset X$ is integral closed of $\delta $-dimension $k$. The groups $Z_ k(X_1)$, $Z_ k(X_2)$, $Z_ k(X_1 \cap X_2)$ are free on the subset of these $Z$ such that $Z \subset X_1$, $Z \subset X_2$, $Z \subset X_1 \cap X_2$. This immediately proves the lemma in this case. The general case is similar and the proof is omitted. $\square$

Lemma 41.12.4. Let $(S, \delta )$ be as in Situation 41.7.1. 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 41.10.3. 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 41.9.1). 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 28.49.1 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 $Y = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$ (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. $\square$


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