Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 126 / 2 / i / Solution

A group scheme over $k$ is a $k$-scheme $G$ equipped with multiplication $m:G{\times }_{k}G\to G$, identity $e:\mathrm{Spec}k\to G$, and inversion $j:G\to G$ satisfying the associativity, identity, and inverse diagrams.

ConsiderThe diagonal morphism is ${\Delta }_{G/k}=q^{-1}(e)$. For a finite-type $k$-scheme the rational identity point is closed, so its inverse image is closed. Thus the diagonal is a closed immersion and every such group scheme over a field is a separated scheme.

In characteristic $p>0$, the infinitesimal additive groupis a nonreduced group scheme. Its comultiplication is $t\mapsto t\otimes 1+1\otimes t$, which is well defined because $(t\otimes 1+1\otimes t{)}^p=t^p\otimes 1+1\otimes t^p$.