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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2011

    I have tried to make the page torsion look more like a disambiguation page and less like a mess. But only partially successful.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 5th 2011

    Heh. I wrote something at the Café, speculating where the term “flat” (as in “flat module”) came from. The wild speculation is that whoever invented the word wanted something that would sound sort like a synonym of “torsionfree” (which for modules over a PID coincides with “flat”), and by a kind of oblique free association, recalled that “torsionfree” for curves in 3\mathbb{R}^3 means “flat” (planar).

    No, I am not confused about the meanings of these terms! I know full well these are very different senses, and I’d have to ascribe a quirky sense of humor to any mathematician who would re-invent the term “flat” based on such a free association. It otherwise means little, and I’m sorry if Urs felt duty-bound to set the record straight (when he surely has more important things to think about).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2011

    Hey Todd,

    you write:

    I’m sorry if Urs felt duty-bound to set the record straight

    no need to apologize! It is true that your comment on the Café made me have a look at our nnLab entry on torsion, but it was not to set straight any record, just to bring that entry into better shape.

    I find your observation quite interesting. The relation of terms that you suggested had not occured to me before. I had thought that, like “torsion”, “flat” is a word that has been re-used for completely independent phenomena again and again. Witnessed for instance by the fact that even if you are indeed right and somebody really meant to refer to differential geometry when coining the word “flat module”, there is then a striking clash with the at least now most common use of “flat = no curvature”!