The Stacks project

43.6 Proper pushforward

Suppose that $f : X \to Y$ is a proper morphism of varieties. Let $Z \subset X$ be a $k$-dimensional closed subvariety. We define $f_*[Z]$ to be $0$ if $\dim (f(Z)) < k$ and $d \cdot [f(Z)]$ if $\dim (f(Z)) = k$ where

\[ d = [\mathbf{C}(Z) : \mathbf{C}(f(Z))] = \deg (Z/f(Z)) \]

is the degree of the dominant morphism $Z \to f(Z)$, see Morphisms, Definition 29.51.8. Let $\alpha = \sum n_ i [Z_ i]$ be a $k$-cycle on $X$. The pushforward of $\alpha $ is the sum $f_* \alpha = \sum n_ i f_*[Z_ i]$ where each $f_*[Z_ i]$ is defined as above. This defines a homomorphism

\[ f_* : Z_ k(X) \longrightarrow Z_ k(Y) \]

See Chow Homology, Section 42.12.


Lemma 43.6.1. Suppose that $f : X \to Y$ is a proper morphism of varieties. Let $\mathcal{F}$ be a coherent sheaf with $\dim (\text{Supp}(\mathcal{F})) \leq k$, then $f_*[\mathcal{F}]_ k = [f_*\mathcal{F}]_ k$. In particular, if $Z \subset X$ is a closed subscheme of dimension $\leq k$, then $f_*[Z]_ k = [f_*\mathcal{O}_ Z]_ k$.

Proof. See Chow Homology, Lemma 42.12.4. $\square$

Lemma 43.6.2. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of varieties. Then $g_* \circ f_* = (g \circ f)_*$ as maps $Z_ k(X) \to Z_ k(Z)$.

Proof. Special case of Chow Homology, Lemma 42.12.2. $\square$

Comments (6)

Comment #3838 by Yuxuan on

You missed a \dim (f(Z)).

Comment #3932 by on

Sorry, I don't understand what you are saying. Please try again.

Comment #4427 by awllower on

In the first paragraph the definition of the pushforward seems to be incomplete and missing something after dim (f(Z)).

Comment #4509 by on

I'm hoping this is fixed now. Keep your fingers crossed...

Comment #5520 by Steve Kudla on

On line 5, a typo, \alpha is a k-cycle on X.

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