# Is taking the product of quasi-projective varieties associative?

I was reading a bit of Hartshorne, and I know that the product of quasi-projective varieties is again a quasi-projective variety.

I should hope that taking products is associative, but I am unsure. What I mean by this is that if $X, Y, Z$ are all quasi-projective varieties over some algebraically closed $k$, is it always true that $(X\times Y)\times Z\cong X\times (Y\times Z)$ so that we can not bother with the parentheses?

I searched around, but everything led me back to 3.16 in Hartshorne.

• Yes. This is true of all categorical products. In fact there is a distinguished such isomorphism. – Qiaochu Yuan Jan 6 '15 at 2:58

It is a fact that $X \times Y$ is a categorical product, meaning that it has a universal property, namely this: it comes equipped with two maps, called the projections, $\pi_1:X \times Y \to X$ and $\pi_2:X \times Y \to Y$ satisfying the following property. For any other object $Z$ and maps $\alpha:Z \to X$ and $\beta:Z \to Y$, there is a unique map $\gamma:Z \to X \times Y$ such that $\pi_1 \circ \gamma = \alpha$ and $\pi_2 \circ \gamma = \beta$.