In commutative diagram we define the term 'flat module', but what is actually the meaning of the adjective 'flat'? I guess that the 'flat' here maybe a geometric description of 'torsion free', which is an equivalent expression in the context of PID. And torsion free has good geometric image. But I cannot persuade myself to accept this description. I would appreciate it if anyone who is familiar with the original usage of this terminology can come to my aid.
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1$\begingroup$ I was under the impression that "flatness" meant similar to Euclidean. Still don't understand this but I think (at least classically) all flat modules are isomorphic to a filtered colimit of free modules $\sum_i R^{s_i}$ I would really appreciate if someone who had a better background could advise. You might also want to look into the history of mathematics site hsm.stackexchange.com $\endgroup$– Molly Stewart-GallusNov 20, 2021 at 18:25
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$\begingroup$ I think this is a natural question and shows sufficient research, so I upvoted. While Math.SE would benefit from a nice Answer, I wonder if you've read the Wikipedia article on this topic? I'm not sure what.aspects of your Question it leaves unanswered. $\endgroup$– hardmathNov 20, 2021 at 18:27
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1$\begingroup$ @hardmath I didn’t see anything on the Wikipedia page why the specific word “flat” might have been chosen by Serre. That is, I think, the thrust of the question. “Flat” has strong geometric associations. $\endgroup$– Thomas AndrewsNov 20, 2021 at 19:18
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1$\begingroup$ @hardmath I think it's a good question but I also think it's probably a duplicate. $\endgroup$– rschwiebNov 21, 2021 at 3:17
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$\begingroup$ @rschwieb: Perhaps you are thinking about this 2014 Question, "What is the intuition behind the name 'Flat modules'?" Serre's paper was in French (and 43 pages long), and the link to an English translation in that older Question is "403 Forbidden". $\endgroup$– hardmathNov 21, 2021 at 4:55
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