
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)) $d = [\mathbf{C}(Z) : \mathbf{C}(f(Z))] = \deg (Z/f(Z))$ is the degree of the dominant morphism$Z \to f(Z)$, see Morphisms, Definition 28.49.8. 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 $f_* : Z_ k(X) \longrightarrow Z_ k(Y)$ See Chow Homology, Section 41.12. Lemma 42.6.1. 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 41.12.3.$\square$Lemma 42.6.2. 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 41.12.2.$\square$## Comments (2) Comment #3838 by Yuxuan on You missed a $after$\dim (f(Z)). Comment #3932 by on Sorry, I don't understand what you are saying. Please try again. ## Post a comment Your email address will not be published. Required fields are marked. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi\$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

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