# Questions tagged [gorenstein]

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### Is a certain ring a Gorenstein ring?

Let $K$ be a local number field, $L$ a quadratic reduced $K$-algebra over $K$, and let $a\in O_L$ and consider $R = O_K[a]$, where $O_K, O_L$ are the ring of integers of $K, L$, respectively. Under ...
• 751
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### For complete local domain of dimension $1$, when does vanishing of $\text{Ext}^1_R(\omega_R, R)$ forces $R$ to be Gorenstein?

Let $R$ be a complete local domain of dimension $1$ with canonical module $\omega_R$. If $\text{Ext}^1_R(\omega_R, R)=0$, then does it follow that $R$ is Gorenstein ? Is it true at least for some more ...
• 493
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• 493
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### A One-Dimensional Local Ring Admitting a Finitely Generated Reflexive Module with Finite Injective Dimension Must Be Gorenstein

Let $(R, \mathfrak m)$ be a one-dimensional commutative Noetherian local ring. Let $M$ be a finitely generated $R$-module with finite injective dimension $\operatorname{injdim}_R(M).$ One can prove ...
• 4,810
1 vote
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### Analytically unramified local ring of dimension $1$ always admit canonical module?

Let $(R, \mathfrak m)$ be a reduced local ring of dimension $1$ such that the completion of $R$ is also reduced (such rings are called analytically unramified). Note that $R$ is Cohen-Macaulay since ...
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• 4,384
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### $R$ Gorenstein implies $\operatorname{Proj}(R)$ Gorenstein

Let $k$ be a field and let R be a Gorenstein $k$-algebra which has a non-negative grading $R=\oplus_{k\geq 0} R_k.$ Assume further that $R_0=k$ and that $R$ is generated in degree one. I've seen it ...
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• 4,384
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### Does the completion of a local Gorenstein ring has finite injective dimension over the original ring?

Let $(R, \mathfrak m)$ be a local Gorenstein ring and $\hat R$ be its $\mathfrak m$-adic completion. So we have a canonical map $R \to \hat R$ which makes $\hat R$ into an $R$-module. My question is: ...
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### Local Cohen-Macaulay ring over which every finitely generated module of finite injective dimension also has finite projective dimension

Let $(R,\mathfrak m,k)$ be a local Cohen-Macaulay ring. If every finitely generated $R$-module that has finite injective dimension also has finite projective dimension, then is it true that $R$ is ...
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• 1,560
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### Local Gorenstein ring such that the square of the maximal ideal contains every minimal prime ideal

Let $(R, \mathfrak m)$ be a commutative local Gorenstein ring such that every minimal prime ideal of $R$ is contained in $\mathfrak m^2$. Then is it true that $R$ is an integral domain ? If this ...
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### Example of Gorenstein ring that has infinite Krull dimension?

Let $R$ be a Noetherian commutative ring. If $R$ is local, then $R$ is Gorenstein if $inj.dim(R)<\infty$. Otherwise $R$ is Gorenstein if $R_P$ Gorenstein for all prime ideals $P$.\ There is a ...
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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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### "Manifolds are not Gorenstein"

I recently heard someone say that manifolds are not Gorenstein. To me being Gorenstein means that the canonical divisor is Cartier. So I don't understand why manifolds aren't Gorenstein, since on a ...
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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 ...
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