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Looking at some French papers, it seems that the word "trait" is often used to refer to the spectrum of a discrete valuation ring $A$.

Does anyone know what the translation of this should be? Is it supposed to be a word conveying some geometric intuition for $\text{Spec}(A)$?

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    $\begingroup$ A link to the relevant papers could be helpful to figure out the actual English equivalent. $\endgroup$
    – Sylvain JULIEN
    Commented Dec 8, 2019 at 18:12
  • $\begingroup$ publications.ias.edu/sites/default/files/Theorie-de-Hodge-I.pdf in Section 8 $\endgroup$
    – Kim
    Commented Dec 8, 2019 at 19:22
  • $\begingroup$ In which page does the word "trait" appear? $\endgroup$
    – Sylvain JULIEN
    Commented Dec 8, 2019 at 19:32
  • $\begingroup$ @SylvainJULIEN Page 428, "Soit $V$ le spectre ... (un trait hensélien)". $\endgroup$
    – Z. M
    Commented Jul 18, 2023 at 6:54

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This means a little segment of line: not a point, but the smallest thing after a point.

This also means "feature", which could be another reason to use this for a spectrum.

See https://en.wiktionary.org/wiki/trait for the etymology.

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    $\begingroup$ It is also the equivalent of "stroke", as in the stroke order while writing Japanese kanji. $\endgroup$
    – Sylvain JULIEN
    Commented Dec 8, 2019 at 18:10
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    $\begingroup$ While the explanation as "the smallest line segment larger than a point" catches the geometry quite clearly, I do not understand your remark about "feature". $\endgroup$
    – Fred Rohrer
    Commented Dec 9, 2019 at 21:12
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Someone means something as $\text{Spec}(k[[x]])$: his spec is $0$ and $(x)$, so it is an enlargement of the point $(0)$ of $\text{Spec}(k)$, for this reason it could be named as a "trait". It is inside the longer line $\text{Spec}(k[x])= \mathbb{A}_k^1$: in this sense you can think at it as the "really little" piece of the affine line around $x=0$.

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It's supposed to be analogous to an open unit disk: $$D = \{|z| < 1\}$$ See:

http://176.58.104.245/NOTES/Dobbiaco-2014-06/Illusie-Dobbiaco.pdf (Page 6)

https://perso.math.univ-toulouse.fr/btoen/files/2015/02/Oxford-clay-2017.pdf (Page 11)

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