# Are the rings $k[[t^3,t^4,t^5]]$ and $k[[t^4,t^5,t^6]]$ Gorenstein? (Matsumura, Exercise 18.8)

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

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.
what is the conductor of $<3,4,5>$ and $<4,5,6>$? – messi Jul 1 '12 at 17:49