# Automorphism group of a scheme, 2

Hi,

I have the following two questions about automorphism groups of schemes. First of all, let $S$ be a scheme, and $S^c$ its set of closed points. What is the connection between $Aut(S)$ and $Aut(S^c)$ ? Secondly, let $S$ be a $\mathbb{Z}$-scheme, and $S_k$ the base-extension to some field k. What is the precise relation between their automorphism groups ? Thanks,

THC

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What do you mean by an automorphism of $S^c$? – Laurent Moret-Bailly Jan 16 '12 at 11:36
Laurent, I interpreted the question asking whether two automorphisms that agree on closed points would agree, so accordingly one could define $Aut(S^c)$ as the image of $Aut(S)$ in the set of all self-maps of $S^c$... – Sándor Kovács Jan 16 '12 at 11:59
I decided to interpret the question in terms of set-theoretic permutations, just in case Sándor's answer wasn't what THC had in mind. – S. Carnahan Jan 16 '12 at 12:16

It depends on what kind of scheme $S$ is. If $S$ is (reduced) of finite type over a field, then the set of closed points is dense in $S$, so all morphisms are determined by their behaviour on the closed points. On the other hand if $S$ is a local scheme, then it has a single closed point, so you can't say much about automorphisms just from the closed points.
The functor from schemes to sets, given by taking the set of closed points, is neither full nor faithful. For example, the spectrum of a field with nontrivial automorphisms has a single closed point, but its automorphism group is nontrivial. On the other hand, the complex projective line has automorphism group $PGL_2(\mathbb{C})$, but the set of closed points is an abstract set of cardinality $2^{\aleph_0}$, and has an automorphism group of strictly larger cardinality. What you can say is that the functor induces a homomorphism (that may be neither injective nor surjective).
If you want to consider automorphism groups of base extensions, you might as well assemble them into the automorphism group sheaf $\underline{\operatorname{Aut}} (S)$, which eats a scheme $T$, and returns the automorphism group of $S \times T$ (as a $T$-scheme). When $S$ is projective, this sheaf is represented by a scheme, whose $k$-points are precisely the automorphisms of $S_k$. For example, $\mathbb{P}^1$ has automorphism group scheme $PGL_2$.