# Who first introduced the notation $\mathcal{O}$ in algebraic geometry or algebraic number theory

This is my first question for HSM. If it is consider too specialized for HSM, perhaps it can be migrated to MathOverflow.

In algebraic number theory, one frequently denotes the ring of algebraic integers in a number field $K$ by $O_K$. Similarly in algebraic geometry, where one studies algebraic varieties and later schemes, it is common to denote the ring of regular functions over an open set $U$ by $O(U)$ or $\mathcal{O}(U)$, and similarly one often denotes the stalk at a point $p$ by something like $O_p$ or $\mathcal{O}_p$.

Where does this $O$ come from, or where was it first used? My best guess is that it originally had something to do with Order in commutative ring theory, starting primarily in algebraic number theory and later spilling over into algebraic geometry.

(I'm asking as a follow-up to a comment I made at MO here, where I explain a private pun I've developed to give a sense to O-notation as it is used in asymptotic analysis; I realized after making that comment that I had no idea where the notation $O$ in the sense of this question comes from.)

• Hi, and welcome to Stack Exchange! I think we would really like to see more high-level (specialized) questions on the site, so I hope yours will receive a satisfactory answer and that you will return again! I've also added some tags to your question; feel free to edit them if you think something is missing (or if I mis-tagged!).
– Danu
Oct 14, 2015 at 20:55
• @Danu Thanks for the welcome! The added tags look appropriate to me, and I think they should suffice (can't really think of others at the moment). Oct 14, 2015 at 21:02
• I'd edit the title. When seeing it I thought it was about the $O(\cdot)$ notation for bounds (as is common in algorithm analysis and elsewhere) Oct 14, 2015 at 21:39
• @vonbrand I changed to a calligraphic O, which should disambiguate from the O notation in asymptotic analysis (but note the last paragraph of my post where I point to a relation between the two). Oct 14, 2015 at 21:48
• Good grief! Why the downvote? Oct 15, 2015 at 5:10

Your guess is right: the notation $\mathfrak o$ goes back to Dedekind. If you get a copy of Dirichlet-Dedekind's Vorlesungen über Zahlentheorie and look in Dedekind's famous XI-th Supplement, which was the first systematic development of algebraic number theory, you'll see $\mathfrak o$ starting in section 170 when Dedekind defines Ordnung (= Order).
• Aha! Thanks very much for this. Can you confirm that the prevalent use of $O$/$\mathcal{O}$ in algebraic geometry is after the fashion in number theory? Oct 15, 2015 at 3:06
• Sure. See math.stackexchange.com/questions/436078/… for a discussion of the transition: Dedekind's $\mathfrak o$ was adopted by van der Waerden in his Moderne Algebra (which was based on lectures of Emmy Noether, who knew Dedekind's work very well, telling others that many basic ideas could be found in it), which in turn inspired Henri Cartan in his notation for rings of holomorphic functions in complex analysis/algebraic geometry.
• On page 62 of Remmert's Theory of Complex Functions, Remmert brings up the notation $\mathcal O(D)$ for holomorphic functions on a suitable domain $D$, which is due to Henri Cartan, and writes about the idea that it honors Oka only to dismiss that by citing a letter he received from Cartan that said his $\mathcal O$ was inspired by its use in van der Waerden's Moderne Algebra.