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I am studying Atiyah and MacDonald's book "Introduction to Commutative Algebra" and I have just read the definition of a flat module.

It seems to me that if they have called that kind of modules "flat" there must be some sort of geometric intuition or something like that. Why are they called so? Or is it just a meaningless name?

Thanks

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    $\begingroup$ Section 1.2 of this article seems to give some geometric intuition for it, although not enough for me to formulate an answer. $\endgroup$ – rschwieb May 2 '14 at 12:42
  • $\begingroup$ Perhaps do you mean "flat" instead of "plane". Eisenbud's book on Commutative Algebra gives some geometric intuition for the term, along with some examples, but like rschwieb's case, not enough for me (I have an awful geometric intuition, tough). $\endgroup$ – Matemáticos Chibchas May 2 '14 at 13:19
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    $\begingroup$ Sorry, I was reading the spanish's version, and translated the term "plano" without too much thinking. I indeed meant "flat modules". I proceed to edit the original message. Thanks $\endgroup$ – Qwertuy May 2 '14 at 13:32
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    $\begingroup$ I have seen that there are some other similar questions. It was my mistake, for not using the right term in English. I apologize $\endgroup$ – Qwertuy May 2 '14 at 13:35
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    $\begingroup$ Here is a good discussion from the pros: mathoverflow.net/questions/6789/why-are-flat-morphisms-flat $\endgroup$ – Adam Saltz May 2 '14 at 13:55
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Flat modules were first defined by Serre in his seminal paper Géometrie Algébrique et Géométrie Analytique (GAGA). (For an English translation, see this link). In that paper, Serre gives no motivation for the name. I actually asked my commutative algebra professor your same question -- why do we call flat modules "flat"? -- and he told me that Serre didn't have a good reason for it. I don't think the name has geometric significance.

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  • $\begingroup$ Given that Serre is still alive, surely someone has asked him. Why isn't there a report of that? $\endgroup$ – Ryan Reich May 2 '14 at 13:56
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    $\begingroup$ @Ryan: Serre doesn't remember! See Brian Conrad's comment in Adam Saltz's link above. $\endgroup$ – Georges Elencwajg May 2 '14 at 14:33

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