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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 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. –  Zev Chonoles Jul 1 '12 at 15:00
    
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. –  messi Jul 1 '12 at 15:08
    
You are right: 1. is not Gorenstein, and 2. is Gorenstein. –  user26857 Jul 1 '12 at 16:29
    
thanks navigetor23, you can write what you wrote above as an answer and i will accept it. This is useful as Matsumura does not give answer to this question. –  messi Jul 1 '12 at 16:45
    
You may also write any other results that enable us to conclude whether $k[[t^{a_1},t^{a_2},t^{a_3}]]$ is gorenstein or not, where $a_i$ are non-negative integers. –  messi Jul 1 '12 at 17:23
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1 Answer 1

up vote 4 down vote accepted

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 $S=<3,4,5>$ is symmetric and this is not the case. On the other side, in the second example $S=<4,5,6>$ is symmetric.

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what is the conductor of $<3,4,5>$ and $<4,5,6>$? –  messi Jul 1 '12 at 17:49
    
For the first the conductor is 3, and for the second it is 8. –  user26857 Jul 1 '12 at 17:50
    
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. –  messi Jul 1 '12 at 17:54
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