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Here is question 18.8 of Matsumura's Commutative Ring Theory. It asks whether the rings

  1. $k[[t^3,t^4,t^5]]$,
  2. $k[[t^4,t^5,t^6]]$

are Gorenstein. I got that 1) is not Gorenstein, but 2) is Gorenstein (by computing the socle). Just wanted to check if I am correct. I don't need the answer necessarily, a yes or a no will suffice. Thanks.

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    You are right: 1. is not Gorenstein, and 2. is Gorenstein.2012-07-01

1 Answers 1

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Let $k$ be a field and $R$ a graded $k$-algebra. Then $R$ is Gorenstein iff $R_m$ is Gorenstein, where $m$ is the irrelevant maximal ideal of $R$. (This is exercise 3.6.20(c) from Bruns & Herzog.)

If $R$ is a Noetherian local ring, then $R$ is Gorenstein iff its completion $\widehat{R}$ is Gorenstein. (This is Proposition 3.1.19(c) from Bruns & Herzog.)

Let $k$ be a field and $S$ a numerical semigroup. Then $k[S]$ is Gorenstein iff $S$ is symmetric. (This is Theorem 4.4.8 from Bruns & Herzog.)

The examples from Matsumura are completions of affine semigroup rings with respect to their irrelevant maximal ideals. For instance, $k[[t^3,t^4,t^5]]$ is Gorenstein iff $k[t^3,t^4,t^5]$ is Gorenstein iff $S=\langle 3,4,5\rangle$ is symmetric and this is not the case. On the other side, in the second example $S=\langle 4,5,6\rangle$ is symmetric.

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    thanks, i will check and see if i get it as 3 and 8 respectively, if not, i might ask for further help. Thanks again, this answer is quite helpful for me.2012-07-01