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.

  • $\begingroup$ Any notation including "$G\to k$" would be misleading. $\endgroup$ – YCor Mar 9 '17 at 21:58
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    $\begingroup$ 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... $\endgroup$ – R. van Dobben de Bruyn Mar 10 '17 at 3:08
  • $\begingroup$ 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 ? $\endgroup$ – thierry stulemeijer Mar 10 '17 at 9:50
  • $\begingroup$ 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. $\endgroup$ – YCor Mar 10 '17 at 13:45
  • $\begingroup$ $\operatorname{Aut}(s)$? $\endgroup$ – 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.

  • $\begingroup$ Thanks for those references ! I could access one of the reference therein, and it is quite interesting. Unfortunately, I could not access the other relevant reference, which is a paper of yours entitles "Abelianization of the second nonabelian Galois cohomology". Would it be possible to provide a copy of this paper ? $\endgroup$ – thierry stulemeijer Mar 12 '17 at 23:33
  • $\begingroup$ Please send me an e-mail, and I will send you the PDF file of the paper. $\endgroup$ – Mikhail Borovoi Mar 13 '17 at 6:07
  • $\begingroup$ See also this preprint and the reference to Florence therein. $\endgroup$ – Mikhail Borovoi Mar 13 '17 at 6:28
  • $\begingroup$ 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. $\endgroup$ – thierry stulemeijer Mar 13 '17 at 10:26
  • $\begingroup$ See also this preprint, Section 1. $\endgroup$ – Mikhail Borovoi Dec 20 '17 at 14:46

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