Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 16 / 2 / ii / Solution

The composite of two finite type morphisms is of finite type: on suitable affine open subsets, take the algebra generators of the first algebra together with the images of the algebra generators of the second. To check separatedness, factor the diagonal morphism of the composite asThe first arrow is a closed immersion, and the second is the base change of the closed immersion ${\Delta }_g:Y\to Y{\times }_{Z}Y$. Hence their composite is a closed immersion.

Finally, after any base change $Z^{\prime }\to Z$, the composite is $X^{\prime }\stackrel{f^{\prime }}{\to }{Y}^{\prime }\stackrel{g^{\prime }}{\to }{Z}^{\prime }$. Both arrows take closed subsets to closed subsets, so their composite does as well. This is universal closedness. All three conditions hold, and therefore