# Questions tagged [gorenstein]

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### Is $R=K[\![x^3,x^2y,xy^2,y^3]\!]$ a Gorenstein ring? [duplicate]

Let $K$ be a field and $R=K[\![x^3,x^2y,xy^2,y^3]\!]$ the ring of formal power series. Is $R$ a Gorenstein ring? $R$ is Cohen-Macaulay of dimension 2. So, I have to check if $Ext^2_{K}(K,R)=K.$
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### On graded Artinian Gorenstein algebras

Let $k$ be a field and $R$ an $\mathbb{N}$-graded $k$-algebra that is graded-commutative. Assume that $\dim_k R<\infty$ and that $R$ is Gorenstein (i.e. the injective dimension of $R$ over itself ...
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### Is the converse of Proposition 3.5.4 (c) of Bruns_Herzog true?

Question 1. Is the converse of Proposition $3.5.4 (c)$ of Bruns_Herzog true? I can see that $R$ is cohen-macaulay. so if one can prove that $r(R)=1$ , $R$ will be Gorenstein. Question 2. ...
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### Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
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### Zero dimensional Gorenstein ring

Let $(R,\mathfrak m)$ be a zero dimensional Gorenstein ring and $\mathfrak q$ be an $\mathfrak m$-primary ideal of $R$. Then TFAE: 1) $\mathfrak q$ is irreducible, 2) $(0:\mathfrak q)$ is principal, ...
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### Example of Gorenstein local ring of dimension 1 [closed]

The ring $k[[x,y]]/(xy)$ is Gorenstein. Why?
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### Gorenstein ring and projective module

I am new to this topic and would appreciate little explanation. Def: A commutative, unital ring $A$ is a cubic ring if $A$ is a free $\mathbb{Z}$-module of rank $3$. Def : A cubic ring $A$ is ...