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I'm studying on "Cohen-Macaulay rings" of Bruns-Herzog, here a link:

http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false

At page 65 there is the Auslander-Buchsbaum_serre theorem, I need help on one implication:

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring then if $\mathrm{proj\;dim}\;k<\infty$ then $R$ is regular. For this implication the book says that it uses the Ferrand-Vasconcelos theorem:

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and $I\neq0$ a proper ideal with $\mathrm{proj\;dim}\;I<\infty$. If $I/I^2$ is a free $R/I$-module, the $I$ is generated by a regular sequence.

I didn't understand how we apply Ferrand-Vasconcelos, could you help me please?

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    @Nicolas: in fact it's what I did, I wrote the implication of Auslander-B-S theorem where I have problem, I wrote the Ferrand-Vasconcelos theorem and I asked how can I apply it to proove the implication of A-B-S that I wrote. The link to the book it's only a help if you want to read the theorem on the book and maybe you understand it better than me2011-06-14

2 Answers 2

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I think I have it. We have that $\mathrm{proj\;dim}\;k<\infty$. Then $k$ has a finite minimal resolution

$0\rightarrow F_s\rightarrow\cdots\rightarrow F_0\rightarrow k\rightarrow0$

But for the costruction of the minimal free resolution we have that $F_0=R\;\;$ and $\;\;\mathrm{Ker}(R\rightarrow k)=\mathfrak{m}$. So $\mathfrak{m}$ has the minimal free resolution

$0\rightarrow F_s\rightarrow\cdots\rightarrow F_1\rightarrow\mathfrak{m}\rightarrow0$.

Am I right?

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Let $I=m$, then the assumptions of F-V is satisfied. So $m$ is generated by a regular sequence...

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    Because proj dim R/m <\infty.2011-07-07