# Why is : $g \circ f = ( \mathrm{pr}_{XZ}^{XYZ} )_* \big( ( \mathrm{pr}_{XZ}^{XYZ} )^* . \mathrm{pr}_{YZ}^{XYZ} )^* g \big)$?

Let $$\mathcal{E} (k)$$ be the enumerative category of smooth projective varieties over a field $$k$$. This is a $$\mathbb{Q}$$ - linear category which has for objects : smooth projective varieties over $$k$$, and for morphismes the algebraic correspondances of degree $$0$$ modulo the numerical equivalence as follows :

$$\mathcal{E} (k) (X,Y) = \{ \ \text{ algebraic cycles ( with rational coefficients ) of codimension } \ \mathrm{dim} X \\ \mathrm{over} \ X \times Y \ \} / \sim_{ \mathrm{num} }$$

In a textbook which i'm learning right now, the author states that :

The composition of morphisms of $$\mathcal{E} (k) (X,Y)$$ is given by the following law : $$g \circ f = ( \mathrm{pr}_{XZ}^{XYZ} )_* \big( ( \mathrm{pr}_{XZ}^{XYZ} )^* . \mathrm{pr}_{YZ}^{XYZ} )^* g \big)$$ Could you tell me why is : $$g \circ f = ( \mathrm{pr}_{XZ}^{XYZ} )_* \big( ( \mathrm{pr}_{XZ}^{XYZ} )^* . \mathrm{pr}_{YZ}^{XYZ} )^* g \big)$$ ?

Thank you.

• Have you tried writing down what you think composition should be? You’ll end up with what is given after some simple manipulations. – Samir Canning Oct 1 '18 at 0:59
• I don't know how to do this. :-) – YoYo12 Oct 1 '18 at 14:30
• First off, there's a typo in your composition formula. Second, have you tried anything? Do you know how a correspondence works? Do you know about the projection formula? – Samir Canning Oct 3 '18 at 2:16