I was wondering if I can get some help understanding a problem.
Namely, consider the Frobenius map of $\mathbb{F}_p$-algebra $f:A \rightarrow A$ given by $a \mapsto a^p$. I showed that if $A$ is an $\mathbb{F}_p$-Hopf algebra then the map $f$ is a homomorphism of Hopf algebras i.e. it is compatible with comultiplication. Now I need to "compute the kernel (as Hopf algebras)", as stated in the problem. What does this mean exactly? Why is this different than just computing the kernel of $\mathbb{F}_p$-algebras?
Further, I need to compute the Frobenius map and it's kernel in the case when $A$ is the Hopf algebra representing the additive group functor $\mathbb{G}_a$. Any hint as to how to go about doing this? I tried again with the requirement that it should be compatible with comultiplication defined on $A$, but I don't know what I am supposed to conclude about the definition of the map $f$ and it's kernel from that.
Let $k$ be the field with $p$ elements, $A$ be the Hopf algebra $k[t]$ with $\Delta(t)=t\otimes1+1\otimes t$, and let $F:A\to A$ be its Frobenius map.
If $B$ is a $k$-algebra, then there is a bijection $\Phi:\hom(A,B)\to B$ (with $\hom$ denoting algebra homomorphisms) given by $\Phi(f)=f(t)$. The comultiplication on $A$ turns $\hom(A,B)$ into a group, and under the bijection $\Phi$ this group corresponds precisely to the additive group $B$.
Now the map $F:A\to A$ gives us a map $f\in\hom(A,B)\mapsto f\circ F\in\hom(A,B)$. Using $\Phi$ to transport this to $B$, you can easily check that that map corresponds to the map $b\in B\mapsto b^p\in B$, which is, as expected, a homomorphism of additive groups. The map induced by $F$ is therefore $b\in B\mapsto b^p\in B$. Its kernel is the subgroup of elements of $B$ whose $p$th power is zero.
It is easy to see that the functor mapping an algebra $B$ to the set of its elements whose $p$th power is zero is represented by the algebra $k[t]/(t^p)$. This algebra has to be a Hopf algebra, as it represents a group values functor. You should find it.