All Questions
Tagged with local-rings cohen-macaulay
25 questions
1
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46
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Are two-periodic two-generated ideals of Gorenstein local domains self-dual?
Let $I$ be a non-principal ideal of a Gorenstein local domain $R$ such that there is an exact sequence $0\to I \to R^{\oplus 2}\to R^{\oplus 2}\to I\to 0$.
Then, must it be true that $I\cong \text{Hom}...
- 1,682
1
vote
1
answer
72
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Cohen--Macaulay rings $R$ of dimension more than $1$ such that $R/\mathfrak p$ is also Cohen--Macaulay for every minimal prime ideal $\mathfrak p$
Let $R$ be a Cohen--Macaulay local ring. If $\dim R=1$, then $R/\mathfrak p$ is also a Cohen--Macaulay ring for every minimal prime ideal $\mathfrak p$ of $R$. It is known that for $\dim R\ge 2$, this ...
- 435
3
votes
1
answer
109
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Finitely generated torsion-free indecomposable modules over one-dimensional complete local domains are isomorphic to ideal?
Let $R$ be a complete local domain of dimension $1$. Let $M$ be a finitely generated torsion-free indecomposable $R$-module. Then, must $M$ be isomorphic to an ideal of $R$?
Also clearly, $R$ embeds ...
- 1,682
3
votes
1
answer
92
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Serre conditions and local isomorphisms
I encountered the following in the introduction of the paper "Duality for Koszul Homology over Gorenstein Rings":
I assume it's easy, but why is this fact true? I played with modding out by ...
- 1,460
1
vote
1
answer
68
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When $R/I \cong S/J$, where $R$ is Cohen-Macaulay, $S$ is regular local and $ht(J)=\mu(J)$ [closed]
Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring. Let $I\subseteq \mathfrak m$ be an ideal of $R$. If $R/I \cong S/J$ for some regular local ring $S$ and ideal $J$ of $S$ such that
$ht(J)=\mu(J)$, ...
- 105
2
votes
1
answer
65
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On local rings $(R, \mathfrak m)$ such that $\text{Spec}(R)$ is disjoint union of $\text{Ass}(R)$ and $ \{\mathfrak m \}$
Let $(R, \mathfrak m)$ be a Noetherian local ring such that $\mathfrak m \notin \text{Ass}(R)$ and $\text{Spec}(R)=\text{Ass}(R)\cup \{\mathfrak m \}$. Then, must $R$ be Cohen-Macaulay?
Of course the ...
- 435
7
votes
0
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213
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Existence of $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal
Let $(R,\mathfrak m)$ be a Noetherian local domain of dimension at least $2$.
Then, must there exist $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal of $R$? What if we also ...
- 105
0
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0
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50
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On maximal Cohen-Macaulay property of a special kind of ideal
Let $R$ be a local Cohen-Macaulay domain of dimension at least $2$. Let $M$ be a maximal Cohen-Macaulay $R$-module such that localization of $M$ at every height $1$ prime ideal of $R$ is free. ...
- 557
0
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1
answer
107
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Does the torsion submodule and the $0$-th local Cohomology module coincide over local Cohen-Macaulay ring?
Let $M$ be a finitely generated module over a local Cohen-Macaulay ring $(R,\mathfrak m)$.
If $x\in M$ is annihilated by a non-zero-divisor $r\in \mathfrak m$ , then is it true that $\mathfrak m^n x=0$...
- 557
1
vote
1
answer
253
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Maximal Cohen-Macaulay modules of full support, over non-artinian local Cohen-Macaulay rings, are faithful?
Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of positive dimension. Let $M$ be a finitely generated maximal Cohen-Macaulay module i.e. $\operatorname{depth} M=\dim R$.
If $\operatorname{Supp}(...
- 3,715
1
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0
answers
129
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Flat extension of local domains
Let $(R,m)$, $(S,n)$ be two local Noetherian domains, $R \subseteq S$ is flat,
and $m \subseteq n$.
Question 1:
If $R$ is regular and $S$ is Cohen-Macaulay,
is $S$ also regular?
Question 2:
If $R$ is ...
- 6,937
0
votes
1
answer
73
views
Ideal $I$ with $\operatorname{depth}(I)=d$ in a local CM ring of dimension $d$
Let $(R,m)$ be a Noetherian Cohen-Macaulay local ring, having Krull dimension $d$
(by this, necessarily $d < \infty$).
Let $I$ be an ideal of $R$ with $\operatorname{depth}(I)=d$,
namely, $I$ ...
- 6,937
4
votes
1
answer
116
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Cohen-Macaulayness and regularity of $A/p$
This question claimed (and proved) that if $p$ is a prime ideal of $A=k[x_1,\ldots,x_n]$
with $\operatorname{ht}(p) \in \{0,1,n-1,n\}$, then $A/p$ is Cohen-Macaulay.
Now, let $A$ be a (Noetherian) UFD ...
- 6,937
2
votes
0
answers
49
views
Local Cohen-Macaulay rings
Case one:
Let $(R,m)$ be a Noetherian local ring of Krull dimension $d$, $\dim(R)=d$.
Let $I$ be an ideal of $R$.
Assume that $\operatorname{depth}(I,R)=d$, namely, the maximal length of a regular ...
- 6,937
0
votes
0
answers
269
views
Flat morphism of local rings
Let $(A,m_A)$ and $(B,m_m)$ be two Noetherian local rings, $A \subseteq B$
and $B$ is a finitely generated $A$-algebra.
Step 1: Assume that:
(1) $A$ is regular.
(2) $A \subseteq B$ is flat.
Question ...
- 6,937
2
votes
1
answer
142
views
Cancelling the canonical module in tensor products
Let $R$ be Cohen-Macaulay local ring with the canonical module $\omega_R$ and let $M$ and $N$ be two finitely generated $R$-modules. Assume that
$$ \omega_R \otimes_R M= \omega_R \otimes_R N $$
Can ...
- 535
1
vote
1
answer
407
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Quotient of a local Cohen-Macaulay ring by a minimal prime
Let $R$ be a local Cohen-Macaulay ring. Let $P$ be a minimal prime ideal of $R$.
Is it true that $\operatorname {depth} R/P=\operatorname {depth} R$ ?
Notice that since $R$ is local Cohen-Macaulay, ...
- 1,682
1
vote
1
answer
355
views
Relationship between the depth of a local Cohen-Macaulay ring and its associated graded ring
For a Noetherian local ring $(R, \mathfrak m)$ , let $\mathrm{gr}_{\mathfrak m} (R):= \oplus_{n \ge 0} \mathfrak m^n/\mathfrak m^{n+1}$ be the associated graded ring.
It is known that $\dim R=\dim ...
- 3,715
2
votes
1
answer
380
views
Local Cohen-Macaulay ring of minimal multiplicity
Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring. How to show that $R$ has minimal multiplicity (i.e. $e(R)=\mu (\mathfrak m)- \dim R +1$ ) if and only if $\mathfrak m^2=(\overline x)\mathfrak m$ ...
- 3,715
3
votes
1
answer
264
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Example of a reflexive canonical module
I want an example of a Cohen-Macaulay local ring $R$ with the canonical module $\omega_R$ such that $\omega_R$ is reflexive (that is $\omega_R \cong {\rm Hom}_R ({\rm Hom}_R(\omega_R,R),R)$).
- 535
3
votes
1
answer
150
views
Surjective homomorphism from a faithfully flat module to a regular local ring.
Let $R$ be a regular local ring and let $M$ be a faithfully flat $R$-module. Does there necessarily exist a surjective $R$-module homomorphism from $M$ to $R$?
For context, I am computing $\sum_{f\in\...
- 2,700
2
votes
0
answers
40
views
Existence of ideal in Cohen-Macaulay ring, going modulo which still gives Cohen-Macaulay [closed]
Let $R$ be a local Cohen-Macaulay ring of dimension $\le 2$. Does there necessarily exist an ideal $J$ of $R$ such that $\sqrt J$ is a minimal prime ideal of $R$ and $R/J$ is Cohen-Macaulay ?
- 3,715
5
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0
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435
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Integral extension of a local ring is semilocal
Let $S\subseteq R$ be commutative rings with $1$. It is given that $S$ is local and $R$ is integral over $S$. I need to show that $R$ is semilocal that is $R$ has finitely many maximal ideals.
It is ...
user612735
3
votes
0
answers
213
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Is there a graded analogy for the dimension criterion for a regular sequence in a C-M local ring?
In the theory of Cohen-Macaulay rings, a basic theorem is that if $(R,\mathfrak{m})$ is a Cohen-Macaulay local noetherian ring, and $x_1,\dots,x_r\in\mathfrak{m}$ is a sequence satisfying $\dim R/(x_1,...
- 21.1k
0
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0
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47
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is there a relationship between $\ell (R/I^n)$ and $\ell (R/I)$
$(R,m)$ is local neotherian cohen-macaulay ring of dimension $d$, and $I$ is an $m$-primary ideal of $R$. since $I$ is an $m$-primary, $\dim R /I=\dim R/I^n =0$. so $\ell(R/I^n)$ and $\ell (R/I)$ are ...
- 7,562