In Marc Levine, Six Lectures on Motives, in the first chapter, the author defines the so-called the category of Chow correspondences. I suffer from certain technical details related to algebraic cycles and correspondences. In the following, $k$ is a field and all schemes are $k$-schemes.
- Let $X$ be a $k$-scheme and $D$ be an its Cartier divisor He claims that we can extend by linear to define an operation $D\cdot\square: z_n(X)_D \to z_{n-1}(D)$ where $z_n(X)_D$ is the free abelian group generated by integral closed subschemes $Z$ of dimension $n$ which is not contained in $D$ and $z_r$ is simply group of $r$-dimensional algebraic cycles, and $D.Z = \sum m_i Z_i$ where $Z_i$'s are irreducible components of $D \times_X Z$, $m_i$'s are certain integers. My question is why do we need the hypothesis $Z \nsubseteq D$ to make the operator $D \cdot \square$ well-defined.
- If $X, Y$ and $Z$ are smooth projective $k$-schemes then we can compose elements in Chow groups (they are also called correspondeces) by the following formula $$\beta \circ \alpha = p_{XZ*}(p_{XY}^*(\alpha) \cdot p_{YZ}^*(\beta))$$ where $\alpha \in CH_{\mathrm{dim}(X)}(X \times Y), \beta \in CH_{\mathrm{dim}(Y)}(Y \times Z)$, $p_{XY}^*,p_{YZ}^*$ are pull-back operators of $$p_{XY}:X \times Y \times Z \to X \times Z, \ p_{YZ}: X \times Y \times X \times Z$$ and $p_{XZ*}$ is push-forward operator of $p_{XZ}:X \times Y \times Z \to X \times Z$. All of $p_{AB}$ defined in the obvious way; being projective make them well-defined. My question is: how can I think about correspondences intuitively? I did search somewhere else but it does not help me at all and correspondences are notoriously abstract that I do not even know how to prove this composition law is associative.