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$(R,m)$ is a local Noetherian ring. $M$ is a nonzero finite $R$-module of finite injective dimension($id$). It is known that if $R$ is Gorenstein, then $M$ has finite flat dimension ($fd$). I wonder if the converse is true? So the question is:

Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is Gorenstein? ($M$ is a non-zero finite $R$-module)

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Yes. As you have stated it, this is a theorem of H.-B. Foxby (Math. Scand. 40 (1977), 5-19, "Isomorphisms between complexes with applications to the homological theory of modules." http://www.mscand.dk/article/download/11671/9687) Actually, Foxby says "projective dimension" instead of "flat dimension", but for finite modules over Noetherian local rings these invariants coincide.

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  • $\begingroup$ In fact, this theorem already appears in the first tome of the Séminaire Samuel Algèbre Commutative (1966/1967) $\endgroup$ – Olivier Feb 16 '15 at 20:16
  • $\begingroup$ That's surprising. On what page? $\endgroup$ – Neil Epstein Feb 16 '15 at 22:25
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Yes, even with no finiteness assumption on $M$. See Bourbaki, Algèbre Commutative X, §8, exercise 8 c).

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  • $\begingroup$ what do you mean by " X, §8, exercise 8 c"? it is till VII $\endgroup$ – user 1 Feb 16 '15 at 12:28
  • $\begingroup$ version: Elements of Mathematics Commutative Algebra Originally published as ELEMENTS DE MATHEMATIQUE, ALGEBRE COMMUTATIVE © 1964, 1965, 1968, 1969 by Hermann, Paris $\endgroup$ – user 1 Feb 16 '15 at 12:31
  • $\begingroup$ Unfortunately they are not translated, but chapter 8 to 10 exist in french and are published by Springer. $\endgroup$ – abx Feb 16 '15 at 12:58

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