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 $f_*[Z]_ k = [f_*\mathcal{O}_ Z]_ k$.

## 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

is the degree of the dominant morphism $Z \to f(Z)$, see Morphisms, Definition 29.49.8. Let $\alpha = \sum n_ i [Z_ i]$ be a $k$-cycle on $Y$. 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

See Chow Homology, Section 42.12.

**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.
$\square$

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