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Questions tagged [cohen-macaulay]

A ring is called Cohen-Macaulay if its depth is equal to its dimension. More generally, a commutative ring is called Cohen-Macaulay if is Noetherian and all of its localizations at prime ideals are Cohen-Macaulay. In geometric terms, a scheme is called Cohen-Macaulay if it is locally Noetherian and its local ring at every point is Cohen–Macaulay.

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The generating set of ideals in regular local ring

Let $R$ be a regular local ring. It is well-known that $R$ is a Cohen-Macaulay ring. Hence $grade(I,R)=ht(I)$ if $I$ is an ideal. If $I$ is a proper ideal, suppose $ht(I)=d$, is there exists $x_1,.....
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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\...
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If $ \operatorname{Ass}(M)= \operatorname{Assh}(M)$, then $M$ is a Cohen-Macaulay $R$-module? [duplicate]

Let $R$ be a local commutative Noetherian ring and $M$ a finitely generated $R$-module. We denote by $ \operatorname{Assh}(M)=\{ \mathfrak{p}\in \operatorname{Ass}(M) \mid \dim R/\mathfrak{p}=\dim M\}...
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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 ?
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Determining maximal Cohen-Macaulay modules over an invariant ring

Suppose that $G$ is a finite small (i.e. reflection-free) subgroup of $\text{GL}(n,\mathbb{C})$ acting on $S := \mathbb{C}[x_1, \dots, x_n]$. Set $R := S^G$. By 5.20 Corollary of this, the maximal ...
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Is a quotient of ring of polynomials Cohen-Macaulay? [closed]

Let $R = \mathbb{R}[x_1 , \ldots , x_n]$ be polynomial ring and $I \subset R$ be a principal ideal with $I = \langle f \rangle$. I know that $R$ is a CM ring. So my question is that: Is the quotient ...
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Background of Commutative Algebra for Cohen-Macaulay orders and bibliography

In this semester, I'm doing a project in Algebra, and I would like take some advices and suggestions. To be more precise, I will study the Cohen-Macaulay Orders and modules in relation to the Krull ...
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Noetherian Catenary ring and Cohen-Macaulay ring

Let $A$ be a Noetherian ring. $A$ is called catenary if for any two prime ideals $p$ and $q$ in $A$, $p\subset q$, every saturated chain of prime ideals starting at $p$ and ending at $q$ have same ...
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Cohen-Macaulay ring without non-trivial idempotent is homomorphic image of Noetherian domain?

Let $R$ be a Cohen-Macaulay ring with no non-trivial idempotent element. Then is it true that there is a Noetherian domain $S$ such that $R\cong S/I$ for some ideal $I$ of $S$ ? If this is not true ...
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For a Cohen-Macaulay local ring grade and height are same

In Matsumura's book ' Commutative ring theory' Theorem 17.4, page 135 its been proved that in a Noetheriam local ring $(A,\mathfrak m)$ for any proper ideal $I$, grade $I=$ ht$I$, where grade of an ...
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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 ...
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Completion and endomorphism ring of injective envelope

Let $(R, m ,k)$ be a commutative Noetherian local ring. We denote by $E$ the injective envelope of $k$ and by $R^~$ the $m$-adic completion of $R$. For any module $M$ over $R$, we let $M^*=Hom_R(M,E)$....
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regular sequences: proving the geometric interpretation

I found the following discussion about the geometric interpretation of regular sequences very helpful: What is a geometric interpretation of regular sequences in various instances? However I tried to ...
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Looking for easy example of 2-dimensional Noetherian domain which is not Cohen-Macaulay

I am looking for an easy example , with proof, of a 2-dimensional Noetherian domain which is not Cohen-Macaulay . I know and it is easy to prove that such an example doesn't exist in dimension 1. ...
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Regular local ring if every maximal Cohen-Macaulay module is free

I have a problem like this "Let $R$ be a Cohen-Macaulay local ring, $\dim R=d$. Given that every maximal Cohen-Macaulay $R$-module is free, prove that $R$ is a regular local ring." My lecturer gave ...
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n-th Syzygies over a Cohen-Macaulay ring are Maximal Cohen-Macaulay for n big enough

I am trying to proof the following: Let $R$ be a Cohen-Macaulay local ring with $\operatorname{dim}(R)=d$ and let $$ 0\to M\to F_{n-1}\to F_{n-2}\to\cdots\to F_1\to F_0 $$ be an exact ...
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A question on finitely generated $k$ algebra

According to this wikipidea link https://en.m.wikipedia.org/wiki/Cohen–Macaulay_ring : Let $R$ be a local ring which is finitely generated as a module over some regular local ring $A$ contained in $R$....
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nonsplit exact sequence for non free module

Let $R$ be a Henselian Cohen-Macaulay (esp. noetherian) local ring and let $M$ be an indecomposable Cohen-Macaulay (esp. finitely generated) $R$-module which is not free. Why can I always find a ...
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h-vector of an unmixed ideal

We know that: if a ring is Cohen-Macaulay, then it is unmixed. But the converse is not true. if a ring is Cohen-Macaulay, then its $h$-vector is positive. The converse is not true. Then I expect to ...
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Structure of Cohen-Macaulay local algebra

I have the following question: Let $(A,\mathfrak m)$ be a Cohen-Macaulay local $k$-algebra, where $A/\mathfrak m=k$. Then there is a homomorphism $R=k[X_1,\cdots ,X_n]\rightarrow S$ so that $S$ is ...
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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 ...
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The natural inclusion $R\varphi\to\mathrm{Hom}_R(C,C')$ induces the above isomorphism modulo $x$.

This is a part in proof of Theorem 3.3.4 in Cohen-Macaulay Rings, Bruns and Herzog. I don't understand it. Here is the Theorem: Theorem 3.3.4. Let $(R,m,k)$ be a Cohen-Macaulay local ring and let $C$ ...
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Cohen-Macaulay Analytic Rings

Let $X \subset \mathbb{C}^n$ be an affine Cohen-Macaulay variety. I would like to know whether for every point $p \in X$, the local ring of germs of analytic functions on $X$ at $p$ is also Cohen-...
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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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Points in linear general position in a section of a curve in $\mathbb P^{n}(\mathbb C)$

Given a natural number $p\in \mathbb N \setminus{0}$, let $X\subseteq\mathbb P^{g+p+1}(\mathbb C)$ be a curve of genus $p$ and degree $2g+p+1$, and $Y\subseteq \mathbb P^{g+p}$ a general hyperplane ...
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Subring of a Cohen-Macaulay ring [closed]

I try to find a obvious of this example: Subring of a Cohen-Macaulay ring is not Cohen-Macaulay. However, I got stuck. Could you help me say it in more detail? Thanks in advanced.
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Embedded Components and Divisors

I'm trying to understand a comment in the introduction to chapter 18 of Eisenbud's Commutative algebra with a View Toward Algebraic Geometry: '..is the unimxedness theorem which explains, for example,...
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Example of a local Cohen-Macaulay domain which is not regular.

If $R$ is regular local ring then $R$ is also Cohen-Macaulay ring and integral domain. I want to show that the converse is false. I come up with this example. Let $K$ be a field and $R=K[X,Y,Z]/(...
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The length of any saturated prime ideal chains between $P$ and $Q$ are the same.

Let $M$ be a finitely generated Cohen-Macaulay $(R,\mathfrak{m})$-module, and $P,Q\in\mathrm{Supp}_{R}(M)$ such that $P\subset Q$. Prove that if the length of any saturated prime ideal chains between $...
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$k[X_1,…,X_n]$ is Cohen-Macaulay (CM) ring

It is normal if we know the ring $k[X_1,...,X_n]$ is Cohen-Macaulay (CM) ring by the definition which is that the ring has unmixed condition. However, I got stuck when I try to prove that ring is CM ...
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Cohen-Macaulay and connected implies equidimensional?

I'm asking for a reality check. It seems to me that since Cohen-Macaulay rings are locally equidimensional, such a ring is either equidimensional or else disconnected (with different dimensions ...
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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,...
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Two dimensional normal rings are Cohen-Macaulay

Let $R$ be a Noetherian normal ring of Krull dimension two, I would like to show $R$ is Cohen-Macaulay without referring to Serre's normality criterion. There is a similar question here but it is not ...
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Zero divisors of a reduced Noetherian ring

My goal is to show that a reduced Noetherian ring of dimension one is Cohen-Macaulay. My question is on why the reduced condition is imposed. Let $ \mathfrak{p} \in \operatorname{Spec}(R) $. If $ \...
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Does a dualizing sheaf $\omega_X$ give rise to a dualizing module?

Let $X = \text{Proj } R$ be a projective equidimensional Cohen-Macaulay scheme, where $R$ is a finitely generated graded Cohen-Macaulay $\mathbb{C}$-algebra and $\mathcal{O}_X(1)$ is ample. Suppose ...
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When is the tensor product of Cohen-Macaulay modules Cohen-Macaulay?

Let $M_1$ and $M_2$ be Cohen-Macaulay modules over a ring $R$. When is $M_1 \otimes_R M_2$ a Cohen-Macaulay module over $R$? That is the finite question I have. Motivation: I know that when $M$ is a ...
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When is the Chow pull-back of a cycle its preimage?

Given a morphism (not necessarily flat) $f \colon X \rightarrow Y$ between smooth varieties over $\mathbb{C}$ say, we have a pull-back homomophism of graded rings $f^* \colon A^*(Y) \rightarrow A^*(X)$...
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translation of “Der kanonische Modul …”

Do you know a note that is the translation of the following in English? J. HERZOG et al., "Der kanonische Modul eines Cohen-Macaulay-Rings," Lecture Notes in Mathematics No. 238, Springer-Verlag,...
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Connection between saturated ideals an CM algebras.

Let $I$ be an homogenous ideal of the polynomial ring $K[x_1,\dots,x_n]$. Is there any relations between $I$ being saturated and $R/I$ being a Cohen-Macaulay?
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When a graded ring is Cohen-Macaulay?

I am trying to solve exercise 19.10 from Eisenbud's Commutative Algebra. I want to show that if $R=k[x_0,...,x_n]/I$ is a graded ring, then $R$ is Cohen-Macaulay iff $R_{\mathfrak p}$ is Cohen-...
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Confusion with arithmetically Cohen-Macaulay varieties

I'm a bit struck about this fact; I think it's really a silly question, but I'm not completely sure about it. Let $X\subseteq \mathbf{P}^m$ be a projective variety; choose the best hypotheses ...
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How to check Cohen-Macaulayness?

Let $R=k[x,y,z]$. Consider the ideal $I=(x^2z^2,xyz,y^2z^4,y^4z^3,x^3y^5,x^4y^3)$. Is $R/I$ Cohen-Macaulay ? By definition it seems tough to solve this problem. Is there any other way to check this?
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Cohen-Macaulayness inherits to quotient (Matsumura, CRT, Exercise 17.4)

A well-known theorem in commutative algebra states the fact that if $R$ is a Cohen-Macaulay ring, and $a_1,...,a_r$ is an $R$-sequence, then $R/I$ is Cohen-Macaulay, where $I=(a_1,...,a_r)$. Now, ...
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A local Cohen-Macaulay ring [closed]

It may be a simple question, but I am stuck at: If $k$ is a field why $k[x^3,x^2y,xy^2,y^3]$ is Cohen-Macaulay when localized at the maximal ideal $(x^3,x^2y,xy^2,y^3)$? Any help? Thanks!
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$R/\operatorname{rad}(0)$ is Cohen-Macaulay

If $(R,m)$ is a local Noetherian ring, is it true that $\bar R=R/\operatorname{rad} (0)$ is a Cohen-Macaulay ring? I think that we should take a maximal chain of prime ideals of length $d$ under $\...
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Examples of Cohen-Macaulay integral domains

Question 1 Could you find a non Cohen-Macaulay ring $A$ without zero divisors. I would like $A$ to be as simple as possible. For instance, I want $A$ to be finitely generated alegbra over $\mathbb{C}$...
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Prove that $ k[x_1,\ldots,x_4]/ \langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle$ is not Cohen-Macaulay.

Prove that $ k[x_1,\ldots,x_4]/ \langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle$ is not Cohen-Macaulay. We have $\langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle=\langle x_1,x_3 \rangle \cap \langle x_2,x_4\...
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Global sections of sheafification of Cohen-Macaulay module

Let $S=k[x_0,\ldots,x_n]$ be the polynomial ring over a field $k$ with the standard grading. Let $M$ be a finitely generated graded Cohen-Macaulay $S$-module of dimension at least two. Let $\mathcal{F}...
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Reduced one-dimensional Noetherian ring is Cohen-Macaulay

If $(R,m)$ is a local Noetherian reduced ring of Krull dimension $1$ then $R$ is Cohen-Macaulay, since in a reduced Noetherian ring the set of zero divisors is the (finite) union $U$ of minimal prime ...
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A Cohen-Macaulay localisation

Let $R=\mathbb C[X,Y]/(Y^3-X^3)$, let $x,y$ be the images of $X,Y$ in $R$, and let $R_1$ be the localization of $R$ at the maximal ideal $(x,y)$. I want to prove that $R_1$ is a Cohen-Macaulay ring. ...