Questions tagged [gorenstein]

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Induced map on the $K_0$ of punctured spectrum of completion

Let $(R,\mathfrak m)$ be a local Gorenstein ring (may also assume excellent) of dimension at least $2$, and let $(\hat R,\hat {\mathfrak m})$ be the $\mathfrak m$-adic completion. Let $U:=\text{Spec}(...
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Notion of simple hypersurface singularity depends on the presentation?

Let $(S, \mathfrak n)$ be a regular local ring. For $0\ne f\in \mathfrak n^2$ define $c(f, S):=\{\text{ideals } I \text{ of } S : f\in I^2\}$ . Now let $(S_1, \mathfrak n_2)$ and $(S_2,\mathfrak n_2)...
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Algebraic characterization (or sufficient condition) when a (graded) local hypersurface has rational singularity

Let $(S, \mathfrak n)$ be a regular local ring of dimension $d\ge 4$ and let $R=S/(f)$ , where $0\ne f \in \mathfrak n^2$. Then $\dim R=d-1\ge 3$. If $\mathfrak m$ is the maximal ideal of $R$ then $\...
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1answer
81 views

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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2answers
111 views

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 ...
3
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1answer
154 views

A local Cohen-Macaulay ring whose dimension is one less than the minimal no. of generators of its maximal ideal

Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring. If $\dim R=\mu (\mathfrak m)-1$ , then is it true that $R \cong S/(f)$ for some regular local ring $S$ and some (non-invertible) regular element $...
3
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1answer
152 views

On relating $\operatorname {Tor}_i^R (k, M)$ and $ \operatorname {Ext}_R^{d-i} (k, M)$

Let $M$ be a finitely generated module of finite projective dimension over a local Gorenstein ring $(R, \mathfrak m,k)$ of dimension ($=$depth) $d$. Then since $R$ is Gorenstein, so $\operatorname {...
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1answer
54 views

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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46 views

On possible closure operation induced by derived functors

For a commutative Noetherian ring $R$, let $ \mathcal I$ denote the set of all ideals of $R$. A function $f : \mathcal I \to \mathcal I $ is called a closure operation on $R$ iff for every ideals $I,...
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1answer
66 views

If $R$ is generically Gorenstein, then $\operatorname{Ann}(\operatorname{Ann}(I))=I$ for every ideal $I$ with non-zero annihilator?

Let $R$ be a Noetherian ring such that $R_P$ is a Gorenstein ring (https://en.wikipedia.org/wiki/Gorenstein_ring) for every minimal prime ideal $P$ of $R$. Is it true that $\operatorname{Ann}_R (\...
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93 views

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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1answer
56 views

$k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements

I have a problem $k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements. I am stuck since I do not have many techniques to prove that an ideal is generated by 2 elements. ...
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63 views

check Gorenstein ring

I have a problem saying "Check that $k[[x,y,z]]/(x^3-z^2,y^2-xz,z^3)$ is a Gorenstein ring while $k[[x,y,z]]/(x,y,z)^2$ is not, given $k$ is a field." Anyway, easily check that Krull dimensions of ...
2
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1answer
164 views

Equivalent definition of maximal Cohen-Macaulay modules over a Gorenstein local ring

$ \newcommand{\Ext}{\mathop{\rm Ext}\nolimits} \newcommand{\depth}{\mathop{\rm depth}\nolimits} \newcommand{\dim}{\mathop{\rm dim}\nolimits} $ A module $M$ is a maximal Cohen-Macaulay ...
2
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1answer
84 views

Is $R=K[|x^3,x^2y,xy^2,y^3|]$ a Gorenstein or a regular ring?

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 $\operatorname{Ext}^2_{K}(...
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70 views

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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1answer
91 views

How to show that CM-finite Gorenstein algebras have finite global dimension?

Let $A$ be a Gorenstein algebra(We call $A$ Gorenstein provided that the regular module $A$ has finite injective dimension on both sides. And if $A$ is Gorenstein, then $inj.dim _AA= inj.dim A_A$ ), $...
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1answer
97 views

When homogeneous coordinate ring of an Abelian variety is Gorenstein?

Let $A$ be an abelian variety of dimension at least $2$. If $A$ is embedded into a projective space by a very ample line bundle $\mathcal{L}$ under which assumptions on $\mathcal{L}$ the homogeneous ...
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84 views

Annihilators of powers of the maximal ideal in an Artinian Gorenstein ring

Let $(R,\mathfrak{m},k)$ be a commutative Artinian Gorenstein ring. Let $n$ be such that $\mathfrak{m}^n\neq0=\mathfrak{m}^{n+1}$. I see that $\mathfrak{m}^n=(0:\mathfrak{m})$, is it true that $\...
2
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1answer
187 views

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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1answer
175 views

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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118 views

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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1answer
354 views

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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1answer
296 views

Example of Gorenstein local ring of dimension 1 [closed]

The ring $k[[x,y]]/(xy)$ is Gorenstein. Why?
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1answer
99 views

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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3answers
459 views

Examples of Noetherian local rings which are not Gorenstein

Can anyone give me an example of a Noetherian local ring which is not a Gorenstein ring?
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1answer
402 views

Matsumura, Exercise 18.8: Cohen-Macaulay and (not) Gorenstein [duplicate]

I need an answer to the exercise 18.8 of Matsumura's book:" Commutative Ring Theory", and generate an algorithm if possible. Let $k$ be a field and $t$ an indeterminate. Consider the subring $A = k[[...
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1answer
189 views

Localization of Gorenstein ring

Let $R$ be a Gorenstein local ring and $S=R \setminus Z(R)$. I want to prove $S^{-1}R =⊕_{ht\ p=0} R_p$ and $S^{-1}R$ is injective $R$-module. I can see the above $p$'s are minimal, $id_{R_p} R_p=0$ ...
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1answer
197 views

Gorenstein, complete intersection

Let $S = k[X_1,...,X_n]$ Example 3.2.11(b) of Bruns-Herzog's book "Cohen-Macaulay Rings", gives a Gorenstein ring that is a complete intersection iff $n \leq 2$. They have proved it as a special case ...
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1answer
251 views

Tor functor for the quotient of a Gorenstein local ring

Let $(R,m)$ be a Gorenstein local ring, $I\subset R$ a perfect ideal of grade $g$ and $S = R/I$. Prove that $S$ is Gorenstein iff $\operatorname{Tor}_g^R(S,S)=S$. This question is Exercise 3.3.25(c) ...
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1answer
560 views

What is a hypersurface ring and why is it Gorenstein?

My question is about this and this: In the first link Graham Leuschke says: "That is a hypersurface ring, so Gorenstein, so the canonical module is the ring itself." What is a hypersurface ...
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248 views

Note or book on Examples of regular, Gorenstein, Cohen Macaulay, … rings

I need a good note or book with plenty of examples in commutative algebra and algebraic geometry which surveyed being regular, Gorenstein, Cohen Macaulay, .... Can you help? thanks.
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1answer
435 views

On Gorenstein ring of dimension zero

Let $R$ be an Artinian local ring. Then $R$ is a Gorenstein ring (i.e., $R$ is an injective $R$-module) iff for any ideal $I$ of $R$, Ann$($Ann$(I))=I$. Why? (We call $R$ Gorenstein if injective ...
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946 views

Gorenstein ring VS. Gorenstein singularity

A normal variety is said to have Gorenstein singularity iff its canonical divisor is a Cartier divisor (one can always define the canonical divisor on a normal variety and it can be proved to be a ...
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285 views

The geometric interpretation of Gorenstein local ring

Many local rings have geometric interpretations. Cohen–Macaulay rings correspond to equi-dimensionality, and regular local rings correspond to non-singularity, but what is a geometric interpretation ...
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1answer
259 views

A Gorenstein domain that is not a complete intersection

Could you give me an example (with proof) of a Gorenstein domain that is not a complete intersection?
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1answer
211 views

The Gorenstein dimension of a ring

I'm studying on these notes. I have a question about page 64, the remark. A local ring is Gorenstein if and only if the Gorenstein dimension of the residue field is finite. Of course if the ring ...
3
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1answer
315 views

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 $k[[t^3,t^4,t^5]]$, $k[[t^4,t^5,t^6]]$ are Gorenstein. I got that 1) is not Gorenstein, but 2) is Gorenstein (...
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2answers
105 views

A non-graded Gorenstein Artin $k$-algebra such that its associated graded ring is also Gorenstein

I am wondering if there is a non-graded Gorenstein Artin $k$-algebra $R$ such that its associated graded ring, $\mathrm{gr}(R)$, is also Gorenstein. All the non-graded Gorenstein rings I tried so ...
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1answer
524 views

Bruns-Herzog problem 3.1.25

This is problem 3.1.25 (page 97) in Cohen-Macaulay Rings by Bruns and Herzog. The direction I am interested in is the following. Let $R$ be a Gorenstein local ring and $M$ a finite $R$-module. If ...
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2answers
337 views

About Gorenstein ring

Is it true that in a (non-local) Gorenstein ring, every maximal ideal has the same height? It seems a little strange, but I don't see any reason why it shoudn't.
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1answer
378 views

Gorenstein ring of dimension zero

Let $(R, \mathfrak m)$ be a local ring and in the same time a finite dimensional algebra over the complex numbers. How one can prove that if $\operatorname{Ann}_R(m)$ has dimension one then $R$ is an ...
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1answer
1k views

Characterization for artinian Gorenstein rings

Let $(R,m)$ be an artinian local ring. Show that if $I \cap J \neq 0$ for all non-zero ideals $I$ and $J$, then $R$ is a Gorenstein ring. Another formulation could be: show that if $(0)$ is an ...
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1answer
353 views

Injective Maximal Cohen-Macaulay modules

Let $R$ be a Gorenstein (not necessarily commutative) ring and let $I$ be an injective finitely generated module over $R$. Is it true that if $\operatorname{Ext}_R^i(I, R)=0$ for $i > 0$, then $I$ ...