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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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    $\begingroup$ You should explain what your reasoning was for getting your answers. That way, people won't tell you things you already know, and if you were confused about something, people will know to explain it. $\endgroup$ Jul 1, 2012 at 15:00
  • $\begingroup$ The above rings are 1 dimensional, so i went modulo a system of parameter and computed the socle, if the socle is a 1 dimensional vectors space then it is Gorenstein. I dont necessarily need the complete solution, basically i am hoping someone could tell me if i am correct or not. $\endgroup$
    – messi
    Jul 1, 2012 at 15:08
  • $\begingroup$ You are right: 1. is not Gorenstein, and 2. is Gorenstein. $\endgroup$
    – user26857
    Jul 1, 2012 at 16:29

1 Answer 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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  • $\begingroup$ what is the conductor of $<3,4,5>$ and $<4,5,6>$? $\endgroup$
    – messi
    Jul 1, 2012 at 17:49
  • $\begingroup$ For the first the conductor is 3, and for the second it is 8. $\endgroup$
    – user26857
    Jul 1, 2012 at 17:50
  • $\begingroup$ 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. $\endgroup$
    – messi
    Jul 1, 2012 at 17:54

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