Questions tagged [gorenstein]

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Module of finite projective dimesion contains a free module

Let $(R, m)$ be a Gorenstein local ring. $M$ is a finitely generated $R$-module with $\dim M=\dim R$ and $\text{pd }M<\infty$. Then $M$ contains a free module $R^k$, such that $$0\longrightarrow R^...
Bromelain's user avatar
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Existence of a proper birational morphism from a Gorenstein scheme, with trivial higher direct images, implies Cohen-Macaulay? [closed]

Let $R$ be a Noetherian excellent reduced local ring containing a field of characteristic $0$. If there exists a Gorenstein scheme $Y$ and a proper birational map $f: Y \to \text{Spec}(R)$ such that $...
Snake Eyes's user avatar
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Examples of Gorenstein local unique factorization domain of dimension $2$ and embedding dimension $4$

I am looking for an example of a local UFD $(R,\mathfrak m)$ of dimension $2$ which is also Gorenstein and $\mathfrak m$ is minimally generated by four elements. Does there exist any such examples?
Snake Eyes's user avatar
1 vote
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Periodicity of the sequence of ideals generared by the entries of the maps in minimal free resolution of modules over complete intersection ring

For a finitely generated module $M$ over a Noetherian local ring $(R,\mathfrak m)$, let $I_i^R(M)$ denote the ideal generated by the entries in a matrix representation of $\partial_i$, where $(F_i,\...
feder's user avatar
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1 answer
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Size of the quotient of a Gorenstein ring with an ideal generated by a regular sequence

Let $(R,\mathfrak{m})$ be a local Gorenstein domain of dimension $2$ with finite residue field, let $x_1,x_2$ be a regular sequence in $R$. Is it true that $R/(x_1,x_2)$ is finite?
Fraz's user avatar
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1 answer
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Is the integral closure of a Gorenstein domain still Gorenstein?

Let $(R,\mathfrak{m})$ be an $n$-dimensional local Gorenstein domain. Is it true that the integral closure of $R$ in its fraction field is still an $n$-dimensional local Gorenstein domain?
Fraz's user avatar
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On $\bigcap_{I \lhd R} (I+\text{ann}_R I)$ in Artinian Gorenstein local ring

Let $(R, \mathfrak m,k)$ be an Artinian Gorenstein local ring. Hence, $\text{ann}_R (\text{ann}_R I)=I$ for every ideal $I$ of $R$. Moreover, $k \cong \text{ann}_R \mathfrak m\subseteq I$ for every ...
uno's user avatar
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Does there exist a Gorenstein local ring $R$ such that $R/\sqrt 0$ is not Cohen-Macaulay?

Does there exist a Gorenstein local ring $R$ such that $R/\sqrt 0$ is not Cohen-Macaulay? Note that since $1$-dimensional reduced rings are Cohen-Macaulay, so for such an example, we must have $\dim R\...
Alex's user avatar
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Any module of grade $g$ can be approximated by a quasi-Gorenstein module of grade $g$?

Let $R$ be a commutative Noetherian ring. For a finitely generated non-zero $R$-module $M$, one defines $$\text{grade}_R(M):=\inf \{j: \text{Ext}^j_R(M,R)\ne 0\} $$ My question is: Is it true that for ...
Alex's user avatar
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Invertibility of trace duals of orders in number fields

Let $K \subseteq L$ be number fields and $S$ be an order in $L$ (not necessarily maximal). Let $R:=S\cap K$ and $S^*$ such as $R^*$ denote the trace duals of $S$ and $R$, respectively. Then $S^*$ and $...
msmo's user avatar
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1 answer
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Localization of Cohen-Macaulay module of finite projective dimension at non-maximal prime ideal

Let $(R,\mathfrak m)$ be a local Gorenstein domain of dimension $2$. Let $M$ be a finitely generated $1$-dimensional module with projective dimension $1$. Then by Auslander-Buchsbaum formula, $\mathrm{...
Snake Eyes's user avatar
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55 views

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 ...
MEEL's user avatar
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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 ...
Snake Eyes's user avatar
1 vote
1 answer
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Is $M\otimes E(R/m)$ an indecomposable module?

In abstract algebra, a module is indecomposable if it is non-zero and can not be written as a direct sum of two non-zero submodules. In other words an $R$-module $M$ is indecomposable, if $M=A\oplus ...
Saeed Yazdani's user avatar
4 votes
1 answer
98 views

Complete local Cohen-Macaulay ring of dimension $1$ whose type equals $1$ less than embedding dimension

Let $(R,\mathfrak m,k)$ be a complete local Cohen-Macaulay ring of dimension $1$. The type of $R$ is then given by $\dim_k \text{Ext}^1_R(k,R)=\mu(\omega)$, where $\omega$ is the canonical module of $...
Snake Eyes's user avatar
2 votes
0 answers
156 views

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 ...
Dylan C. Beck's user avatar
1 vote
1 answer
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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 ...
uno's user avatar
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0 answers
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Colon property in artinian local Gorenstein

I have this problem about a property of Gorenstein artinian rings: Let $(A,m)$ be an artinian local Gorenstein graded ring such that $A_s\neq 0$ and $A_{s+1}=0$ where $A_i$ is the degree $i$ part of $...
Menezio's user avatar
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1 vote
1 answer
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Second syzygy modules over rings , satisfying $(S_1)$ and $(G_0)$ , are reflexive

Let $(R,\mathfrak m,k)$ be a Noetherian local ring such that $R_P$ is Gorenstein for every minimal prime ideal $P$ of $R$ and $\text{depth }R_P\ge 1$ whenever $ht (P)\ge 1$. If $0\ne M$ is a finitely ...
user521337's user avatar
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4 votes
1 answer
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Proving a duality between Ext and Tor for maximal Cohen-Macaulay modules over Gorenstein ring

Let $(R,\mathfrak m, k)$ be a local complete Gorenstein ring of dimension $d$. Let $M,N$ are maximal Cohen-Macaulay modules (i.e. have depth equal to $d$) that are locally free on the punctured ...
user521337's user avatar
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8 votes
1 answer
747 views

What is "self-dual" about Gorenstein rings?

The wikipedia article on Gorenstein rings says In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring $R$ with finite injective dimension as an $R$-module. There are ...
Matt's user avatar
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1 answer
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Relationship Between the Dimension of an $R / \mathfrak m$-Vector Space and Its Completion in the $\mathfrak m$-adic Topology

Consider a Noetherian local ring $(R, \mathfrak m, k).$ We will denote by $\widehat -$ the completion of $-$ with respect to the $\mathfrak m$-adic topology, i.e., the topology on $R$ in which a ...
Dylan C. Beck's user avatar
3 votes
0 answers
61 views

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 $U:=\text{Spec}(...
user's user avatar
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4 votes
1 answer
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$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 ...
mathdonkey's user avatar
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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)...
user521337's user avatar
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0 votes
0 answers
92 views

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 $\...
user's user avatar
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0 votes
1 answer
198 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: ...
uno's user avatar
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6 votes
2 answers
310 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 ...
uno's user avatar
  • 1,560
4 votes
1 answer
247 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 $...
uno's user avatar
  • 1,560
4 votes
1 answer
462 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 {...
uno's user avatar
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1 vote
1 answer
82 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 ...
user's user avatar
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2 votes
0 answers
52 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,...
uno's user avatar
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1 vote
1 answer
135 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 (\...
user521337's user avatar
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2 votes
0 answers
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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 ...
T C's user avatar
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1 vote
1 answer
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Characterisation of Gorenstein Curve

Let $C$ be a curve (so a $1$-dimensional, proper $k$-scheme) and $\omega_C$ it's dualizing sheaf. A curve is called Gorenstein iff $\omega_C$ is an invertible sheaf on $C$, in other words $\omega_C \...
user267839's user avatar
0 votes
1 answer
68 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. ...
T C's user avatar
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0 votes
0 answers
122 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 ...
T C's user avatar
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4 votes
1 answer
547 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 ...
Miriam's user avatar
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4 votes
1 answer
236 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}(...
Problemsolving's user avatar
4 votes
0 answers
90 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 $Ext^2_{K}(K,R)=K.$
Problemsolving's user avatar
3 votes
0 answers
97 views

"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 ...
user2520938's user avatar
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0 votes
1 answer
157 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 $inj.dim _AA= inj.dim A_A$ ), $...
Xiaosong Peng's user avatar
1 vote
1 answer
154 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 ...
Alex's user avatar
  • 6,264
1 vote
0 answers
153 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 $\...
user113589's user avatar
2 votes
1 answer
263 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 ...
user339616's user avatar
3 votes
1 answer
190 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. ...
user 1's user avatar
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5 votes
0 answers
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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 ...
rstarmidi's user avatar
0 votes
1 answer
573 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, ...
jessie's user avatar
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-1 votes
1 answer
424 views

Example of Gorenstein local ring of dimension 1 [closed]

The ring $k[[x,y]]/(xy)$ is Gorenstein. Why?
Martia's user avatar
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1 vote
1 answer
127 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 ...
user124471's user avatar