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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.

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  • $\begingroup$ Have you tried writing down what you think composition should be? You’ll end up with what is given after some simple manipulations. $\endgroup$ – Samir Canning Oct 1 '18 at 0:59
  • $\begingroup$ I don't know how to do this. :-) $\endgroup$ – YoYo12 Oct 1 '18 at 14:30
  • $\begingroup$ 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? $\endgroup$ – Samir Canning Oct 3 '18 at 2:16

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