Dear everyone,

(i) Who is the father of the adjective “syntomic” in algebraic geometry?

(ii) And why did he choose to introduce a new term for what we already know from EGA IV.19.3.6 and SGA 6.VIII.1.1 as “flat, locally of finite presentation, and local complete intersection”?


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    Hi Thanos, Fontaine and Messing claim that Barry Mazur invented the term "syntomic." I'm not sure where the name comes from, but it is a hell of a lot catchier than "flat, locally of finite presentation, and locally a complete intersection!" – Clark Barwick May 15 '10 at 2:28
  • Thanks, Clark! You make a reasonable point, but my hope for (ii) would be an answer that also explains why Mazur chose the striking word “syntomic”, which means “cut short, abridged” in Greek. But now that we know the answer to (i), I'll email Mazur directly. – Thanos D. Papaïoannou May 15 '10 at 3:02
  • I think the idea is that (possibly with some stretch of interpretation) it is supposed to mean complete cut (tomic, as also in cyclotomic, meaning cut), i.e., complete intersection. I have also heard Messing attribute it to Mazur. As to why the answer should be obvious (as Clark pointed out), the "flat, locally of finite presentations and local complete intersection topology" is not something you would want to use more than once in a lecture (if that). – Torsten Ekedahl May 15 '10 at 4:47
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    Ha!, here's a formal improvisation: perhaps Mazur passed from “complete intersection” to the portmanteau “co-section”, which he then translated piece-by-piece to Greek to make “syn-tome”, whose associated adjective is then “syn-tomic”. – Thanos D. Papaïoannou May 15 '10 at 5:04
up vote 16 down vote accepted

Mazur gives the following beautiful justification, which explains the “syn-” in “syntomic” as well.

Dear Thanos,

Thanks for your question. I'm thinking of “ local complete intersection” as being a way of cutting out a (sub-) space from an ambient surrounding space; the fact that it is flat over the parameter space means that each such "cutting" as you move along the parameter space, is---more or less---cut out similarly. I'm also thinking of the word "syntomic" as built from the verb temnein (i.e., to cut) and the prefix "syn" which I take in the sense of "same" or "together". So I think it fits.

Best wishes,


  • Excellent! If it is allowed you should mark this the answer... – Torsten Ekedahl May 16 '10 at 5:38
  • Torsten, apparently it is allowed, after you wait a fixed amount of hours, but it could be bad form. I will go ahead and approve my own answer, but would appreciate comments from the MO censors on whether it's good form or not. – Thanos D. Papaïoannou May 17 '10 at 0:35
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    Well, in this case this is the answer (from the horse's mouth so to speak) and you needed to post the question in order to find out that the concept was coined by Mazur so I don't see how it could be bad form. – Torsten Ekedahl May 17 '10 at 4:09

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