# Notation for the automorphisms of a $S$-scheme over automorphisms of $S$

Here is a slightly anecdotical notational question.

Let $S$ be a scheme and let $X$ be a scheme over $S$, with structural morphism $s\colon X\to S$. Is there a good suggestive notation for the group $\lbrace (f,g)\in \mathrm{Aut}(X)\times \mathrm{Aut}(S)~\vert~sf=gs\rbrace$ ?

After chatting with a categorical friend, this can be described succinctly as the automorphism group of $X\to S$ (considered as an object in the arrow category of schemes).

In fact, I'm interested in finding a good notation when $S$ is (the spectrum of) a field $k$ and $X$ is an affine algebraic group $G$ over $k$. I thought about using $\mathrm{Aut}(G\to k)$ or $\mathrm{Aut}(G\to \mathrm{Spec}(k))$, but there might be something more adapted or already existing in the literature.

• Any notation including "$G\to k$" would be misleading. – YCor Mar 9 '17 at 21:58
• You could also write $\operatorname{Aut}_{\operatorname{\underline{Sch}}^{\to}}(G \to \operatorname{Spec} k)$, but it might not be sufficiently standard that people immediately recognise it... – R. van Dobben de Bruyn Mar 10 '17 at 3:08
• Thank you for your comments. @YCor , I really don't get why a notation including "$G\to k$" would be misleading. Is a notation including "$G\to \mathrm{Spec}(k)$" less misleading ? – thierry stulemeijer Mar 10 '17 at 9:50
• No, $G\to Spec(k)$ is fine. $G\to k$ sounds like whatever kind of arrow from $G$ to $k$, which is not the case. – YCor Mar 10 '17 at 13:45
• $\operatorname{Aut}(s)$? – Thomas Poguntke Mar 11 '17 at 0:56

Indeed, this already exists in the literature. The automorphisms of $G\to\mathrm{Spec}(k)$ are called semilinear automorphisms of $G$, or just semi-automorphisms of $G$, and the corresponding group is denoted by $\mathrm{SAut}(G)$. See Subsection 3.2 of this paper and references therein.
• That's very kind to provide so many interesting references, I've also sent you a mail to obtain your paper. But I now realize that all those authors are in the situation where $k$ is a separably closed field $k_s$, and they only consider semilinear automorphisms over an automorphism of $k_s$ that fix an algebraic subextension, whereas I definitely don't want those restrictions. Well, I guess that if I insist on this difference, it wouldn't be bad to reuse that terminology. – thierry stulemeijer Mar 13 '17 at 10:26