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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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:...
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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 ...
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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 ...
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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, $M$ ...
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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 ...
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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 $...
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Does the torsion submodule and the $0$-th local Cohomology module coincide over local Cohen-Macaulay ring? [closed]

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 $\mathfrak m^n x=0$...
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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. ...
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71 views

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 $\mathfrak S(M)$ ...
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111 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}(...
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On $\operatorname{Hom}_R(S,\omega_R)$ where $S$ is regular domain and finite extension of $R$

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring admitting a canonical module $\omega_R$. Let $R \subseteq S$ be a module finite extension such that $S$ is a regular integral domain i.e. $S$ is an ...
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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 $...
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1answer
101 views

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 ...
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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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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 ...
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equality of support of a module and its quotient

Reference: Atiyah and Macdonald, Introduction to Commutative Algebra, page 46. Let $A$ be a commutative ring with 1 and $M$ a $A$-module. Then we define $\mathrm{Supp}(M)$ to be the set of prime ...
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Torsion-free Cohen-Macaulay modules

We say an $R$-module $M$ over integral domain $R$ is a torsion-free module if zero is the only element annihilated by some non-zero element of the ring $R$. Let $R=K[[x_1,\dots ,x_d]]$, $d>1$, be ...
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Minimal prime ideals of Cohen-Macaulay modules of positive dimension are minimal primes of the ring?

Let $(R, \mathfrak m)$ be a local ring of dimension $d>0$. Let $M$ be a finitely generated Cohen-Macaulay $R$-module (i.e., $\operatorname{depth}M=\dim M$). Then each localisation of $M$ is also ...
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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 ...
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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, $I$ ...
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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 ...
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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 ...
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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 ...
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Does taking integral closure and completion commute?

For a Commutative ring $R$ with total ring of fractions ( https://en.m.wikipedia.org/wiki/Total_ring_of_fractions) $K$, let $\overline R$ denote the integral closure of $R$ in $K$. Let $(R,\mathfrak m)...
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Dualizing module implies Cohen-Macaulay

For a local noetherian ring $(R,\underline{m},k)$, $\gamma$ is a dualizing module iff $\gamma$ is finitely generated and has finite injective dimension over $R$ $\mathrm{Hom}_R(\gamma,\gamma)\cong ...
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$1$-dimensional local Cohen--Macaulay ring whose normalisation is a module finite extension [closed]

Let $(R,\mathfrak m)$ be a local $1$-dimendlsional Cohen-Macaulay ring with total ring of fractions (https://en.m.wikipedia.org/wiki/Total_ring_of_fractions) $Q(R)$. Let $\bar R$ be the integral ...
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Irreducibility of a morphism and indecomposability of a module in a minimal MCM approximation

I've been self-studying Cohen-Macaulay rings using the book by Leuschke-Weigand, and had a question which didn't seem to be answered in the book, but where the results seem to be used implicitly. My ...
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Finite birational extension of $1$-dimensional Cohen-Macaulay ring

Let $R$ be a Cohen-Macaulay ring of dimension $1$ with total ring of fractions $Q(R)$. Let $R\subseteq S \subseteq Q(R)$ be a ring such that $Q(S)=Q(R)$ and $S$ is a (module) finite extension of $R$. ...
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A case when $I \otimes_R M$ and $ IM$ are locally isomorphic at minimal primes?

Let $(R,\mathfrak m)$ be a local Cohen--Macaulay ring of dimension $1$. Let $M$ be a finitely generated $R$-module of depth $1$. Let $I$ be an ideal of $R$ of height $1$. Then my question is: Is it ...
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Vanishing of certain Ext groups for some two-dimensional singularities

Let $R = \mathbb{C}[x,y]$ (or possibly $R = \mathbb{C}[[x,y]]$, although I think we don't need the local hypothesis, and can work in a graded setting instead) and let $G \leqslant \operatorname{GL}(2,\...
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Checking a ring is not Cohen-Macaulay

While reading a book, I found an example that said that the ring $K[w,x,y,z]/(wy,wz,xy,xz)$ is not Cohen-Macaulay. In order to check this, it is stated to take the quotient by the ideal generated by ...
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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 ...
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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 ...
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If $I$ is generated by a regular sequence, $\bigoplus_{n \geq 0}I^{n}/I^{n+1}$ is isomorphic to a polynomial ring

Let $R$ be a Cohen-Macaulay ring and $I$ be an ideal generated by a regular sequence. I want to show that: $\bigoplus_{n \geq 0}I^{n}/I^{n+1}$ is isomorphic to a polynomial ring over $R/I$ in as many ...
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Divisor class group of $2$-dimensional excellent, local normal rational singularity

Let $(R, \mathfrak m)$ be an excellent, local normal domain of dimension $2$ (hence Cohen-Macaulay) with an algebraically closed residue field $k=R/\mathfrak m$. Assume that $IJ$ is an integrally ...
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1answer
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Divisor class group of $2$-dimensional surface isolated singularity

For $n\ge 2$, consider the $2$-dimensional Noetherian local ring $A_n:=k[[x,y,z]]/(x^2+y^2+z^n)$ , where $k$ is an algebraically closed field of Characteristic zero. I can show that each $A_n$ is an ...
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AF+BG theorem and Cohen Macaulay property

I tried to solve the following exercise in Vakil's notes Question 1: According to the hint, I should try to show the intersection of affine cones is CM (since one dimensional scheme is CM iff it has ...
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1answer
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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 ...
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On the colon ideal of a torsion-free module inside it's reflexive hull

Let $(R,\mathfrak m)$ be a Noetherian local complete domain of dimension $1$, with fraction field $K$. (all these assumptions on $R$ imply in particular that for every finitely generated $R$-algebra ...
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1answer
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On the length of $M^{**}/M$ for a finitely generated module on a dimension $1$ complete local domain.

Let $M$ be a finitely generated, non-zero , torsion-free module over a complete local $1$-dimensional Noetherian domain $(R,\mathfrak m)$. Let $n=\mu(M)=l_R(M/\mathfrak mM)$ Then $n\ge r:=\dim_K M\...
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usual method of argument in the linkage theory

Suppose that R is a CM local ring, and I and J are two ideals of R. We say that I is algebraically linked to J if there is an R-sequence x in $I \cap J$ such that I = x : J and J = x : I. In some ...
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Why general elements of ideal are non-zero divisors?

I'm readind the article: A formula for the core of an ideal, by Bernd Ulrich and Claudia Polini and I'm having trouble each time the doubt involves the concept of general element of an ideal Well, ...
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Examples about Cohen-Macaulay property of rings and book recommendation on intuition

My professor asks me to give an example about a local non-CM rings that are CM after modding out at any minimal prime. After a while failing to do so, I believe it is impossible since there exists a ...
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Dualizing complex of Cohen-Macaulay variety

I have a question on a proposition from Shihoko Ishii's book "introduction to Singularities": Preliminaries: A Noetherian local ring $R$ with the maximal ideal $m$ is called Cohen-Macaulay ring if $\...
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1answer
136 views

Well behaved colon between powers of ideals when the associated graded ring is Cohen-Macaulay.

I'm reading a paper: A formula for the core of an ideal, by Claudia Polini and Bernd Ulrich and I'm in trouble with the following problem: Let $R$ be a Cohen-Macaulay ring and $I$ be an ideal of $R$...
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1answer
120 views

On a particular kind of simplicial complex with maximum facet size of $3$.

Let $\Delta$ be an abstract simplicial complex on $n$ vertices such that $\max \{|F| : F$ is a face of $\Delta \}=3$. Let $f_2$ be the number of faces of size (cardinality) $3$ and $f_1$ be the ...
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Hilbert-Samuel multiplicity of standard graded $k$-algebra which is an integral domain and $k$ is algebraically closed

Let $R=\oplus_{i\ge 0} R_i $ be a graded domain such that $R_0=k$ is an algebraically closed field, $R$ is finitely generated $k$-algebra and $R=k[R_1]$. Let $d=\dim R>0$. Let $\mathfrak m=\oplus_{...
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1answer
56 views

On embedding $\mathbb C[x_1,...,x_d]/P $ inside $\mathbb C[[T]]$

Let $P$ be a prime ideal of $\mathbb C[x_1,...,x_d]$ such that ht$(P)=d-1$ i.e. $\dim (\mathbb C[x_1,...,x_d]/P)=1$. Then is it necessarily true that there exists an injective $\mathbb C$-algebra ...
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1answer
170 views

Regular sequence in Cohen-Macaulay ring is regular on Maximal CM-module

I saw an exercise which says the following: For a Cohen-Macaulay ring $R$ and a maximal Cohen-Macaulay module $M$, an $R$-regular sequence is also $M$-regular. Why is this true? The only thing I ...
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1answer
127 views

On the intersection of integral closures of all powers of maximal ideal

Let $(R,\mathfrak m)$ be a Noetherian local ring of positive dimension. For an ideal $J$ of $R$, let $\bar J$ denote the integral closure of $J$ (https://en.m.wikipedia.org/wiki/...