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).

  • $\begingroup$ My guess. From the symbol for holomorphic functions. $\endgroup$
    – OR.
    Commented Jul 4, 2013 at 13:41
  • $\begingroup$ And where does that come from? $\endgroup$
    – Dominik
    Commented Jul 4, 2013 at 13:44
  • $\begingroup$ See mathoverflow.net/questions/92135/… $\endgroup$
    – bradhd
    Commented Jul 4, 2013 at 14:02
  • $\begingroup$ @Brad I think this answers my question. Could you turn that into an answer? $\endgroup$
    – Dominik
    Commented Jul 4, 2013 at 14:10

1 Answer 1


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)"

  • 1
    $\begingroup$ And why did van der Waerden use it? $\endgroup$
    – Dominik
    Commented Jul 4, 2013 at 14:18
  • 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. $\endgroup$ Commented Jul 4, 2013 at 14:21
  • 4
    $\begingroup$ I just learned from Keith Conrad at HSM that your speculation about order is correct: hsm.stackexchange.com/questions/2922/… $\endgroup$
    – user43208
    Commented Oct 15, 2015 at 4:01
  • $\begingroup$ For those who are curious, van der Waerden Moderne Algebra, 2nd German edition, §16, begins with "let $\mathfrak{o}$ be a ring". $\endgroup$ Commented Nov 28, 2022 at 16:17

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