Sheafs are often denoted by the letter $\mathcal O$. What does this O stand for? To me it seems that more natural choices of symbols for sheaves would be $\mathcal S$ or $\mathcal F$ (for the french faisceau).
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$\begingroup$ My guess. From the symbol for holomorphic functions. $\endgroup$– OR.Jul 4, 2013 at 13:41
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$\begingroup$ And where does that come from? $\endgroup$– DominikJul 4, 2013 at 13:44
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$\begingroup$ See mathoverflow.net/questions/92135/… $\endgroup$– bradhdJul 4, 2013 at 14:02
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$\begingroup$ @Brad I think this answers my question. Could you turn that into an answer? $\endgroup$– DominikJul 4, 2013 at 14:10
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1 Answer
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Quoting the historical footnote in Grauert/Remmert, Coherent Analytic Sheaves, concerning the origin of the notation $\mathcal{O}$ for the rings of holomorphic functions (and their associated sheaves):
Some people think the symbol $\mathcal{O}$ was chosen in honor of Oka, sometimes it is even said that $\mathcal{O}$ reflects the French pronunciation of holomorphe. The truth is that the symbol was chosen accidentally. In a letter to the authors from March 22, 1982, H. Cartan writes: "Je m'étais simplement inspiré d'une notation utilisée par van der Waerden dans son classique traité 'Moderne Algebra' (cf. par exemple §16 de la 2e édition allemande, p.52)"
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2$\begingroup$ Presumably because Dedekind used $\mathfrak{o}$ for an order ("Ordnung" in German). But that may also be idle speculation, I don't know why. $\endgroup$ Jul 4, 2013 at 14:21
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4$\begingroup$ I just learned from Keith Conrad at HSM that your speculation about order is correct: hsm.stackexchange.com/questions/2922/… $\endgroup$ Oct 15, 2015 at 4:01
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$\begingroup$ For those who are curious, van der Waerden Moderne Algebra, 2nd German edition, §16, begins with "let $\mathfrak{o}$ be a ring". $\endgroup$ Nov 28, 2022 at 16:17