Relative center of relative group scheme

Question

This might be an easy question. Let $p: X \rightarrow S$ be a relative group scheme. In particular the fibers are group schemes. I want to know if there are constructions like the ``relative center'' or ''relative automorphism group'' and how one would define them in a good way.

Answer 1

There are two problems. One is to define these objects first as a functor, and then to investigate whether the functors are representable. This answer is about the first problem.

This is dealt with in SGA 3 Exposé I 2.1-2.3 from the point of view of group-objects in the category of set-valued contravariant functors on an arbitrary category. For the case of group schemes over a base $S$ this would be functors on the category of schemes over $S$ with values in groups.

Automorphism functors can then be defined in a straightforward way as invertible morphisms from an object to itself in the same category. One can also define group actions, centralizers and normalizers of actions, and finally the center of a group as the centralizer of the action of the group on itself by conjugation.

For automorphisms, giving an element of $\text{Aut}_S(G)(T)$, for an $S$-scheme $T$, comes down to giving a family of automorphisms of the group $\text{Hom}(T',G)$ for every scheme $T'$ over $T$, in such a way that is compatible with the transition maps induced by morphisms $T''\rightarrow T'$ of schemes over $T$. This is the same as giving an automorphism of the functor $G$ restricted to the category of schemes over $T$.

Similarly, a functorial point $\text{Cent}(G)(T)$ is an element of the group $\text{Hom}_S(T,G)$ such that conjugation by every pullback of it via a map $T' \rightarrow T$ of $S$-schemes induces the identity on $\text{Hom}_{T}(T',G_T)$.

These are the correct ways to define these objects. Whether they are representable as (group) schemes is another matter, and I don't know the current state of the art. Hopefully someone more knowledgeable will complete this answer.