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Created Henselian pair and linked to it from Henselian ring and Hensel’s lemma. Probably the definition could be given in more generality for topological rings. The existing definition seems to implicitly assume the discrete topology, since in that case, restricted formal power series are just polynomials, and that’s what appears in the definition I copied over from the Stacks Project.
So, what will be the recommendation on the capitalization question ? So far the entry has “henselian pair” but also “Henselization”.
Is there a consensus among mathematicians who write about such things?
My mistake on the capitalisation, I’m not sure if there is a consensus. I would personally copy what the Stacks Project does (lowercase), if people don’t mind, but there may be a reasonably standard reference (EGA?, Hartshorne?) that has a different stance. Wikipedia sticks with uppercase.
Does Wikipedia have a style guide? I’m thinking that they probably write ’abelian’ in many articles, but also ’Noetherian’ in others. Which is hard to justify on any principled basis.
Roughly speaking, I’d say that adjectives named after authors tend to retain capitals at least for a while (e.g., ’Pynchonesque’, ’Shakespearean’), but by the time we get to verb forms it changes to uncaps (’bowdlerize’, ’boycott’). Thus I’d be inclined to go with ’Henselian’ and ’henselization’. Of course, my language is American English; it seems to me that on the European Continent there is more and more a tendency to use lower case, even for the name itself. (I haven’t done statistics or n-grams on this.)
In mathematics my experience is that a given person’s name is usually capitalized, but a few people are honored by non-capitalization, regardless of part of speech. E.g. we say both “abelian” and “abelianize”, and category theorists say “cartesian” (although other mathematicians often still capitalize it), but we also have for instance “De Morgan algebra” and “De Morganization”.
For reference, an MSE thread about lowercased names in math.
I think the issue is that in French, they don’t capitalize names when they are used as adjectives. So if a term originated in French, there’s some question of whether you anglicize the capitalization convention. On the Stacks project, the conventions seems to be to capitalize.
Anyway, I just added a characterization of Henselian rings that I recently learned from Akhil Mathew, which implies that the property of $(R,I)$ being Henselian depends only on the ideal $I$, viewed as a non-unital ring (except for the basic requirement that $I$ is contained in the Jacobson radical of $R$).
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