Is this true that finitely generated flat module over an integral domain is projective. If Yes, please provide a proof.
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$\begingroup$ This follows immediately from Endo's theorem (i.e. a finitely generated flat module over a commutative ring is projective iff its tensor product with the total ring of fractions is projective). $\endgroup$– tj_Commented Aug 16, 2023 at 19:23
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$\begingroup$ @tj_ Endo, "On flat modules over commutative rings" (1962) refers to Cartier's 1958 paper for the requested result, in his first paragraph. In other words, he is generalising Cartier's statement. $\endgroup$– Dave BensonCommented Aug 16, 2023 at 20:01
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$\begingroup$ @Dave: Yes, of course, one can directly refer to Cartier's paper. I personally prefer English references, because more people are familiar with English than with French. $\endgroup$– tj_Commented Aug 16, 2023 at 23:21
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Yes, this is true. Lemma 5 on page 249 of Cartier, "Questions de rationalité des diviseurs en géométrie algébrique" says that over an integral domain, whether or not it is Noetherian, if the localisation of a finitely generated module at every maximal ideal is free, then the module is projective. Over a local ring, every finitely generated flat module is free (see for example Matsumura, "Commutative Ring Theory" Theorem 7.10). Putting these together gives you a positive answer to your question.
answered Aug 16, 2023 at 16:59