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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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 ...
uno's user avatar
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Going mod an element which annihilates a prime ideal (Associated primes)

Suppose $R$ is a Cohen-Macaulay local ring and $P=ann(x)$, an associated prime of $R$. Now consider the ring $S=R/(x)$. Will $PS$ (extension of $P$ to $S$) still consists of zero divisors of $S$? Or ...
dongrugose's user avatar
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Tensor product of field with Noetherian ring is Cohen-Macaulay

Let $R$ be a Noetherian ring, free as $\mathbb{Z}$-module. Suppose $R \otimes K$ is Cohen-Macaulay for some field $K$ of characteristic $p > 0$. Show that $R \otimes L$ is Cohen-Macaulay for every ...
Anik Bhowmick's user avatar
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Homomorphisms between regular local rings with regular system of parameters

Let $(R,\mathfrak{m})$ be a regular local ring with regular system of parameters $x_1, . . . , x_n$. Let $f : R \rightarrow S$ be a homomorphism of local rings, $f_i = f(x_i)$ for $i = 1, . . . , n$, ...
Anik Bhowmick's user avatar
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Embedding of modules of finite length

Let $R$ be a Cohen-Macaulay ring of dimension $n$. Let $M$ be a finitely generated Artinian $R$-module. One can choose an $R$-sequence $(x_1,...,x_n)$ such that there exists a short exact sequence $0\...
Bromelain's user avatar
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3 votes
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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 ...
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Example of noetherian local ring without Maximal Cohen-Macaulay modules.

A finitely generated module $M$ with $\mathrm{depth}_{R}(M)= \dim R$ is called Maximal Cohen-Macaulay module. I want to find an example of noetherian local ring without Maximal Cohen-Macaulay modules. ...
Well's user avatar
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Non-toric ring which is Cohen-Macaulay

I know a lot of examples of classes of binomial ideals $I$ in $S=K[x_1,\dots,x_n]$ whose $S/I$ is a Cohen-Macaulay domain. Basically, if $I$ is a toric ideal and there exists a monomial order $<$ ...
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If a resolution $f:Y\to X$ satisfies $R^if_*\omega_Y=0$ and $R^if_*\mathcal{O}_Y=0$ for all $i>0$, then do we have $f_*\omega_Y=\omega_X$?

Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic (but I'm mainly interested in positive characteristic). Assume that $X$ has rational singularities,...
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Connection between heights of $(U:_R E)$ and $(ann(E):_R ann(U))$ for torsion-free modules $U\subseteq E$ of constant rank

Let $R$ be a Noetherian local ring. Let $E$ be a finitely generated torsion-free $R$-module of constant rank $e$. Let $s$ be an integer such that $s\geq e+1$. Let $U$ be an $R$-submodule of $E$ and ...
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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 $...
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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 $ht(J)=\mu(J)$, ...
feder's user avatar
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Cohen-Macaulay property of graded ideals

Let $R=K[x_1,\dots,x_n]$ and $I$ be a graded ideal of $R$. My question is the following: if $R/I$ is a Cohen-Macaulay ring then $I$ is a Cohen-Macaulay ideal?
Hola's user avatar
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When is a Noetherian Standard graded algebra over a field Cohen-Macaulay? Any counter-example when it is not the case?

Let $R$ be a Noetherian standard graded algebra with $R_0 = k$, a field, then it is finitely generated over $R_0$ by $R_1$ and is the homomorphic image of some $k[x_1, \ldots, x_n]$, hence isomorphic ...
metalder9's user avatar
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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 ...
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The definition of Cohen-Macaulay algebra

Let $x=(x_1,\ ...\ ,x_d)$ a system of parameters of local ring $(R,m)$. We know that an $R$-algebra $B$ is called Cohen-Macaulay algebra if $x_{i+1}$ is a non-zero divisor in $(x_1,\ ...\ ,x_i)B$ for ...
tensor's user avatar
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Module-finite extensions and reflexive modules.

$\textbf{Question:}$ Let $R\rightarrow S$ be a module-finite extension of complete local rings, with $S$ Regular. Prove that if $M$ is a reflexive $R-$module such that ${\rm Ext}^i_R(M^*,S) = 0$ for $...
Well's user avatar
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1 answer
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Proposition 3.3.3 in Bruns and Herzog

The proposition says that If $(R,m,k)$ is a Cohen-Macaulay (CM) local ring of dimension $d$ and $C$ is a maximal Cohen-Macaulay (MCM) $R$-Module, then a) Suppose $M$ is a MCM $R$-module with $Ext^j_R(...
Naba Kumar Bhattacharya's user avatar
7 votes
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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 ...
feder's user avatar
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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. ...
Snake Eyes's user avatar
1 vote
2 answers
197 views

Non Cohen-Macaulay rings of dimension 2

I am working in the context of Noetherian commutative rings with unit. It is well-known that for a field $K$, the ring $R=K[x^4,x^3y,xy^3,y^4]$ is a 2-dimensional standard graded $K$-algebra which not ...
walkar'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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How small can $\dim(R/P)+\text{ht}(P)$ be?

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d$. Since $\dim(R/P)+\text{ht}(P)\le d=\dim(R/\mathfrak m)+\text{ht}(\mathfrak m)$ holds for every prime ideal $P$ of $R$, so $$\sup \{ \...
Alex's user avatar
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finitely generated contracted ideal

Let $f:R\to T$ be pure ring homomorphism of Noetherian commutative ring i.e.;(for every R-module $M$ we have $f\otimes id_M:R\otimes M\to T\otimes M$ injective) and $a$ is proper ideal of $R$. Let $a^...
pink floyd's user avatar
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4 votes
1 answer
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Regulare sequences and tensor product of modules

Let $(R,\mathfrak m)$ be a Noetherian local ring of Krull dimension $d$ and $M$,$N$ two maximal Cohen-Macaulay $R$-modules,that is Cohen-Macaulay with their dimensions equal to dimension of $R$. ...
Saeed Yazdani's user avatar
2 votes
1 answer
218 views

If $R/I$ is Cohen-Macaulay, is $R/I^2$ also Cohen-Macaulay?

Let $R$ be a local ring and $I \subset R$ an ideal such that $R/I$ is cohen-macaulay. Is $R/I^2$ also cohen-macaulay? I've seen that it always works for principal ideals here: Is a quotient of ring of ...
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Ext and Krull dimension

Let $R$ be a commutative Noetherian ring, $I$ is a proper ideal of $R$ and $M$ is f.g. $R$-module. I want to find an example such that if $c:=\sup\{i\in \mathbb{N_0} \mid \textrm{Ext}_R^i(R/I,M)\neq ...
pink floyd's user avatar
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Grade and projdim of finite modules

I am reading Cohen-Macaulay Rings by Bruns & Herzog, and trying to prove the lemma below: Lemma. Let $R$ be a Noetherian ring, and let $M$ be a finitely generated $R$-module. Then for all $\...
K. Y.'s user avatar
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grade and minimal generator of ideal and zero-divisor

Let $R$ be a Noetherian ring (not necessarily local) and $I$ be the proper ideal of $R$, and $M$ be a finitely generated module such that $grade(I, M) = n > 1$, and $I$ has a generating set $\{x_1,....
pink floyd's user avatar
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0 votes
1 answer
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Tensor product of maximal Cohen-Macaulay modules

Let $(R,\frak{m})$ be a commutative Cohen-Macaulay local ring and $M$ and $N$ two maximal Cohen-Macaulay modules of Krull dimension $d$. Is $M\otimes_RN$ Cohen-Macaulay?
Saeed Yazdani's user avatar
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Maximal Cohen-Macaulay of finite flat dimension

Let $(R,\frak{m})$ be a commutative Cohen-Macaulay local ring and $M$ a maximal Cohen-Macaulay module of Krull dimension $d$. If $M$ has a finite flat dimension, is it true to say that its injective ...
Saeed Yazdani's user avatar
8 votes
2 answers
215 views

Example that M* is not reflexive

Let $R$ be a noetherian ring. Set $(-)^\ast={\rm Hom}_R(-,R)$. For each $R$-module $N$, let $\pi_N:N\rightarrow N^{\ast\ast}$ be the map which maps $n\in N$ to $(f\mapsto f(n))$. $N$ is called ...
Jian's user avatar
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-2 votes
1 answer
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Question about grade of an ideal

Let $K$ be a field and $S=K[[X,Y,Z,W]]$. Consider the elements $f:=XW-YZ$, $g:=Y^3-X^2Z$ and $h:=Z^3-Y^2W$ of $S$ and set $R=S/\langle f\rangle$ and $I:=\langle f,g,h\rangle /f$. I know that $\mathrm{...
pink floyd's user avatar
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-1 votes
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Maximal regular sequence coincides with system of parameters

I'm needing help in this question. Let $k$ be a field. Consider the $k$-algebra $R:=k[x,y,z,w]/(z+w,xy+xw)$ and define the ring $A$ the localization of $R$ in its maximal ideal $\mathfrak{m} = (\...
Kevin Vasconcellos's user avatar
1 vote
1 answer
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Proposition 3.6 of Yoshino's book--Characterization of maximal Cohen-Macaulay modules

I am reading Chapter 3 of Yoshino's book. There is a proposition that the author didn't prove. It looks interesing. But I don't know how to prove. The proposition is, Let $(R,\mathfrak{m})$ be a ...
Jian's user avatar
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1 vote
0 answers
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Associated primes and Cohen-Macaulay rings

In the book Cohen-Macaulay rings by Bruns and Herzog. Part of theorem 2.1.2 states Let $(R,m)$ be a Noetherian local ring, and $M\neq0$ a Cohen-Macaulay $R$-module. Then $\dim R/p=\operatorname{depth}...
user782932's user avatar
1 vote
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
2 votes
0 answers
63 views

On the submodule $(0:_Ma)$ of a non Cohen-Macaulay module $M$

Let $(R,m)$ denote a commutative Noetherian local ring and M a finitely generated R-module of dimension $d$. We say M is a Cohen-Macaulay R-module, if $dim_RM=depth_RM=d$. For every $r\in R$ let $(0:...
Saeed Yazdani's user avatar
4 votes
0 answers
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Is $(\omega:_{Q(R)}\omega)=R$ for a canonical ideal $\omega$ of a one-dimensional local Cohen-Macaulay ring?

Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring of dimension $1$ admitting a canonical ideal $\omega\subseteq R$. Then $\omega$ contains a non-zero divisor. Let $Q(R)$ be the total ring of ...
Simron 's user avatar
1 vote
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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
6 votes
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When $R/\operatorname{Ann}_R(M)$ is a Cohen-Macaulay ring?

Let $(R,\frak m)$ denote a commutative Noetherian local ring and $M$ a finitely generated $R$-module. We say $R$ is Cohen-Macaulay, provided $\operatorname{dim}R=\operatorname{depth}R$. Similarly, $M$ ...
Saeed Yazdani's user avatar
1 vote
1 answer
117 views

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
101 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
0 votes
1 answer
94 views

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 $\mathfrak m^n x=0$...
Snake Eyes's user avatar
1 vote
1 answer
139 views

Bruns & Herzog 1.4.9: Rank of projective modules

I'm reading Cohen-Macaulay Rings by Bruns & Herzog. The lemma below which is used in the proof of the Proposition 1.4.9 and whose proof is left to the reader is difficult for me to prove: Lemma. ...
K. Y.'s user avatar
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1 answer
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Partial order on the set of non-split short exact sequences ending at a MCM module

Let $R$ be a Henselian Cohen-Macaulay (CM) local ring with maximal ideal $\mathfrak m_R$. For an indecomposable finitely generated maximal Cohen-Macaulay (MCM) $R$-module $M$, the set $\mathfrak S(M)$ ...
user771160's user avatar
1 vote
1 answer
220 views

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}(...
user521337's user avatar
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2 votes
1 answer
310 views

Completion (as a module) of the integral closure of a Cohen-Macaulay local ring whose completion is reduced

Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring. Let $\overline R$ be the integral closure of $R$ in the total ring of fractions of $R$. Let $\widehat R$ be the $\mathfrak m$-adic completion of $...
Louis 's user avatar
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2 votes
1 answer
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Example of non-reduced Noetherian dimension 1 Cohen-Macaulay ring.

As the name suggested, is there a non-reduced Noetherian dimension 1 Cohen-Macaulay ring? I know that all reduced Noetherian ring with dimension 1 is Cohen-Macaulay, however it seems difficult for me ...
user124697's user avatar
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3 votes
0 answers
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What are the prerequisites to study Cohen-Macaulay?

I am doing a reading in advanced algebra and I need some help in Cohen-Macaulay : My doubts are What are the prerequisites I need to know before knowing what is Cohen-Macaulay? (I know about basic ...
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