Coordinate Hopf algebras for multiplicative type

For an affine $k$-group scheme $G$, its coordinate algebra $A$ has maps $\mathrm{\Delta }:A\to A{\otimes }_{k}A$, $ϵ:A\to k$, and $S:A\to A$ obtained by reversing multiplication, identity, and inverse. A commutative Hopf $k$-algebra means a commutative unital algebra with these algebra maps satisfying coassociativity, counit, and inverse identities. The inverse identity is $m(S\otimes 1)\mathrm{\Delta }=m(1\otimes S)\mathrm{\Delta }=\eta ϵ$, where $m$ is algebra multiplication and $\eta :k\to A$ is the unit. A Hopf map respects all three maps. A group-like element is $a$ with $\mathrm{\Delta }(a)=a\otimes a$ and $ϵ(a)=1$.

Group schemes and their morphisms have the convention of Group schemes of finite type over a field. In particular the term group here refers to the whole scheme, including nilpotents, rather than only its points over a field.