Let $R$ be a possibly noncommutative left Noetherian ring and $M$ an $R$-module. I am looking for a reference or a proof for the following fact: $M$ is finitely generated and projective if and only if it is finitely presented and flat. (I am not interested in references that treat only the commutative case.)
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4$\begingroup$ The proof given in stacks.math.columbia.edu/tag/058Q works for the general case. $\endgroup$– Mariano Suárez-ÁlvarezCommented Jun 20, 2014 at 4:53
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3 Answers
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This holds over any ring, noetherian or not. See Bourbaki Algebra X, §1, no. 5.
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You can find a detailed proof in Chapter 2, §4D of T. Y. Lam, Lectures on modules and rings, Grad. Texts in Math., Springer, 1999. (More precisely, it follows from Proposition 4.26 a), Proposition 4.29 and Theorem 4.30.)
answered Jun 20, 2014 at 4:46
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An even more general statement is proved in Borceux and Quinteiro's Enriched Accessible Categories.
They show that the Cauchy completion of a small $\mathcal{V}$-category $\mathcal{C}$ is equivalent to the category of $\mathcal{V}$-finitely presentable $\mathcal{V}$-flat presheaves on $\mathcal{C}$.
To get the result you ask about, take $\mathcal{V} = \textbf{Ab}$ (the category of abelian groups) and treat a ring $R$ as a $\mathcal{V}$-enriched category $\mathcal{R}$ with one object. Then use the fact that the Cauchy completion of $\mathcal{R}$ is the category of finitely generated projective $R$ modules.
