# Questions tagged [cohen-macaulay-rings]

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72
questions

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### Quotient Cohen-Macaulay ring and associated primes

Does there exist such (local) Cohen-Macaulay Noetherian ring $(R,m)$ with some $p\in \operatorname{Ass}R$ so that $R/p$ is not Cohen-Macaulay? (I am especially curious about the case where $R$ is a ...

1
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0
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73
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### Grade is not equal to injective dimension

Let $R$ be commutative Noetherian ring but not necessary local ring, and $I$ be proper ideal of $R$. I want to find an example of ring such that
$\operatorname{Ext}_R^i(R/I,R)\neq 0$ is not zero at ...

7
votes

1
answer

341
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### Gorenstein varieties: why the two definitions are equivalent?

There are two definitions of Gorenstein singularities
in the literature. Using Grothendieck's (or Serre's) duality, one
defines the "dualizing sheaf" an object $\hat K_M$ of derived category
...

3
votes

0
answers

100
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### Tensor product by the canonical module preserves Cohen-Macaulayness

Let $X$ be a $\mathbb{Q}$-Gorenstein variety of dimension at least $2$. Suppose that $X$ is normal and Cohen-Macaulay with at worst isolated singularities. Let $F$ be a maximal Cohen-Macaulay $\...

2
votes

2
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246
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### Why is $M$ torsion-free?

I am studying the following article
https://www.math.nagoya-u.ac.jp/~takahashi/tc9.pdf
The main theorem is the Theorem 3.3. Howewer, I have the following questions about the proof:
How does it help ...

1
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0
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84
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### Cohen-Macaulay coordinate rings defined by regular sequences

Consider the polynomial ring $R = k[x_1, \ldots, x_n]$ in $n$ indeterminates over an algebraically closed field $k$ (my relevant case is the complex numbers). Furthermore, consider an algebraic ...

0
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0
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82
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### Complete local Noetherian ring whose normalized dualizing complex has depth equal to dimension of the ring

Let $(R, \mathfrak m,k)$ be a complete Noetherian local ring. Let $D$ be a normalized dualizing complex of $R$. If $\text{depth}_R D=\dim R$, then must $R$ be Cohen-Macaulay (i.e. must $D$ be actually ...

9
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1
answer

233
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### Is the affine closure of the basic affine space of a reductive algebraic group Cohen–Macaulay?

Let $G$ be a reductive algebraic group with choices of Borel subgroup and maximal torus $B \supseteq T$ and unipotent radical $U$ over an algebraically closed field $k$ of characteristic zero. Is the ...

4
votes

2
answers

284
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### Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$

Let $(R,\mathfrak m,k)$ be an Artinian Gorenstein local ring such that $$\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$
Then, is it true that every non-maximal ...

2
votes

0
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147
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### When is a maximal Cohen-Macaulay module of finite homological dimension?

Let $R$ be a local Cohen-Macaulay Noetherian ring. A maximal Cohen-Macaulay module or mCM-module is an $R$-module $M$ of finite type such that $\text{dim }M = \text{depth }M =d$
A module $M$ is of ...

5
votes

2
answers

454
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### question about commutative diagram in category theory

I am reading the article
Maurice Auslander, Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Colloque en l'honneur de Pierre Samuel (Orsay 21-22 mai 1987), ...

3
votes

2
answers

134
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### For a Cohen-Macaulay module $M$ of dimension $t$ over a local CM ring of dimension $n$, is $\text{Ext}^{n-t}_R(M,\omega)$ Cohen-Macaulay?

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $n$ with a canonical module $\omega$. Let $M$ be a finitely generated $R$-module with $\text{depth } M=\dim M=t$. Using Bruns&...

1
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0
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103
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### $A \to B$ with $A$ regular imply that $B$ is CM

The answer to this question says the following:
"The general statement is if $A \to B$ is finite and injective, and $A$ is noetherian and regular, then $B$ is CM if and only if $A \to B$ is flat.
...

1
vote

1
answer

180
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### Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular

Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:
(1) $R$ and $S$ are (Noetherian) integral domains.
(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...

2
votes

1
answer

171
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### On the functors $\text{Hom}_R(k,-)$ and $k \otimes_R ( -)$ for Artinian local Gorenstein ring $R$

Let $(R, \mathfrak m,k)$ be an Artinian local Gorenstein ring, hence $\text{Hom}_R(k, R)\cong k$, and so
$\text{Hom}_R(k, R^{\oplus n})\cong k^{\oplus n} \cong k \otimes_R R^{\oplus n} , \forall n \ge ...

8
votes

1
answer

216
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### Class group of hypersurfaces of finite representation type

Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...

1
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1
answer

138
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### Quotient of ideal generated by regular sequence is a perfect module

I have recently started reading Bruns-Herzog's 'Cohen Macaulay rings' and this is problem 1.4.27 in it.
We say that a module $M$ over a Noetherian ring $R$ is perfect if the projective dimension of $M$...

11
votes

1
answer

332
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### Fixed point scheme of finite group Cohen-Macaulay?

Let $X$ be a quasi-projective scheme over a field $k$.
Let $G$ be a finite group acting on $X$ whose order is invertible in $k$.
If $X$ is Cohen-Macaulay, can we conclude that the subscheme of fixed ...

4
votes

1
answer

141
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### Weighted blowup of a Cohen-Macaulay ring along a regular sequence is Cohen-Macaulay?

Let $(R, \mathfrak{m})$ be a Cohen-Macaulay ring, let $f_1, \dotsc, f_d \in \mathfrak{m}$ be a regular sequence, and let $n_1, \dotsc, n_d > 0$ be weights (feel free to assume that $n_1 = 1$ if it ...

5
votes

2
answers

210
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### Linkage and Cohen-Macaulay-ness

Suppose I have a reduced l.c.i. scheme with two irreducible components: $X = Y \cup Z$. I want to say that if $Y$ is Cohen-Macaulay then $Z$ is as well.
I think this follows from Eisenbund Theorem 21....

5
votes

0
answers

122
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### On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory.
Allow me to first give a minor introduction.
Let $(...

5
votes

0
answers

90
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### Can the conductor of a local, unramified, Cohen-Macaulay domain ever be contained in a parameter ideal?

Let $(R, \mathfrak m)$ be a local Cohen-Macaulay domain which is analytically unramified (i.e. the $\mathfrak m$-adic completion of $R$ is reduced). Let $\bar R$ be the integral closure of $R$ in the ...

3
votes

1
answer

232
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### Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface

Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) .
Also assume that $-1$ is not a square in $k$ . Consider the homogeneous ...

2
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0
answers

120
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### Dualizing complex description in Stacks project

The question is closely related to this one this one (more precisely the reference the comment by AGl earner) and is aimed to understand the proof of Lemma 20.2 from notes from Stacks notes from ...

5
votes

0
answers

163
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### Pushforward of a maximal Cohen-Macaulay sheaf

Let $X$ be an (affine) noetherian, Cohen-Macaulay scheme of dimension at least $2$. Choose a closed point $x \in X$. Denote by $U:=X\setminus \{x\}$ the open subset and $i:U \to X$ the natural ...

3
votes

0
answers

114
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### Is a Maximal Cohen-Macaulay sheaf on a Cohen-Macaulay scheme locally free?

Let $F$ be a Maximal Cohen-Macaulay module on a Cohen-Macaulay scheme $X$.
By definition $\mathrm{depth}_{\mathcal{O}_{X,x}}(F_x)=\mathrm{dim}(\mathcal{O}_{X,x})$ and $\mathrm{depth}(\mathcal{O}_{X,x})...

9
votes

1
answer

421
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### Equivalence of definitions of Cohen-Macaulay type

I know that the Cohen-Macaulay type has these two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring and $M$ a finite $R$-module of depth $t$. The number $r(M) = \dim_k ...

4
votes

1
answer

433
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### difference between Cohen Macaulay and locally Cohen Macaulay curve

I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves.
The stacks project https://stacks.math.columbia.edu/tag/02IN says that a scheme (a curve in ...

4
votes

1
answer

249
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### For every prime ideal $P$ of any Cohen-Macaulay ring $R$, is the sequence $\operatorname{depth}(R/P^n)$ eventually constant?

Let $P$ be a prime ideal of a Cohen-Macaulay ring $R$. Then is the sequence $\operatorname{depth}(R/P^n)$ eventually constant ?

3
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0
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157
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### On Cohen-Macaulay schemes

I know that the irreducible components of a Cohen-Macaulay (CM) scheme don't need to be Cohen-Macaulay. But what happens if we assume that that the irreducible components are CM, what can we say about ...

7
votes

1
answer

574
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### Depth under localization over a Cohen-Macaulay ring

Let $A$ be a Cohen-Macaulay local ring, and let $M$ be an $A$-module that is (S$_2$) and has depth $\ge n$ for some fixed $n$. Let $\mathfrak{p} \subset A$ be a prime of height $\ge n$. Is it true ...

8
votes

1
answer

156
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### For isolated singularity algebra, is every maximal Cohen-Macaulay module locally projective?

Let $R$ be a Cohen-Macaulay noetherian local ring. Let $\Lambda$ be a noetherian $R$-algebra which is maximal Cohen-Macaulay as an $R$-module, where for every nonmaximal prime $\mathfrak{p}$, $\...

4
votes

1
answer

161
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### Cohen-Macaulay rings in GIT

I'm starting to learn about Cohen-Macaulay rings in my commutative Algebra course, and my teacher mentioned its use in Geometric Invariant Theory. Having some basic knowledge on the subject and ...

7
votes

0
answers

258
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### Hironaka decomposition over $\mathbb{Z}$?

Let $A=\bigoplus_{\ell\geq 0}A_\ell$ be a finitely generated graded $\mathbb{Z}$-algebra, with $A_0=\mathbb{Z}$, that is free as a $\mathbb{Z}$-module.
If $k$ is a field, then $A\otimes k$ is a ...

5
votes

1
answer

195
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### If a finite poset supports a Cohen-Macaulay ASL, how far can it be from Cohen-Macaulay?

By the fundamental work of De Concini, Eisenbud, and Procesi, an algebra with straightening law (ASL) must be Cohen-Macaulay if it is built on a Cohen-Macaulay poset. I would like to understand the ...

2
votes

1
answer

466
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### Dualizing sheaf on a Cohen-Macaulay scheme

I would like to fill a detail in the answer given by Francesco Polizzi to the question "Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?".
Let $X$ be a normal, Cohen-Macaulay scheme of ...

10
votes

2
answers

1k
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### Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?

I am reading the paper Frobenius splitting of Hilbert schemes
of points on surfaces by Kumar and Thomsen. At the end of Lemma 11, they seem to imply that the dualizing sheaf on a Cohen-Macaulay scheme ...

2
votes

1
answer

265
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### Does the graded face poset of a BN-pair admit a EL-labeling?

Let $G$ be a finite group with a BN-pair of rank $n$. Let $B$ be the associated Borel subgroup.
Let $P$ be the poset of proper right cosets (i.e. $Kg$ with $K \in [B,G)$ and $g \in G$).
Let $\hat{P}:...

5
votes

0
answers

217
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### symbolic powers of prime ideals

Let $R$ be a CM ring and $P$ be a prime ideal. Let $P^{(n)}$ denote the $n$th symbolic power of $P$.
Is the sequence $\operatorname {depth} (R/P^{(n)})$ eventually constant?

2
votes

1
answer

566
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### The concept "opposite" of Cohen-Macaulayness

Let $S = \mathbb k[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb k$ and let $I \subseteq S$ be a monomial ideal.
For a monomial ideal $J$, let $\#(J)$ be the smallest number of ...

2
votes

0
answers

86
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### Is the bounded coset poset of a boolean interval of finite groups, Cohen-Macaulay?

Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above ...

3
votes

1
answer

334
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### Cohen-Macaulay non-normal toric variety

Given a quasi-smooth toric variety $X$ in the sense of Gelfand-Kapranov-Zelevinsky,
i.e. a (not necessarily normal) toric variety $X$ whose normalization is a $\mathbb{Q}$-factorial toric variety and ...

6
votes

2
answers

638
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### Auslander-Reiten theory for Gorenstein algebras

In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...

1
vote

0
answers

150
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### Björner-Wachs theorem for posets admitting an EL-labeling

In the survey paper Poset Topology: Tools and Applications by Michelle Wachs, there is the following theorem on p46:
Theorem 3.2.4 (Björner and Wachs [40]). Suppose $P$ is a poset for
which $\...

2
votes

1
answer

190
views

### when there is an injection $0 \to R \to K_R$?

Let $(R,m)$ be a Cohen-Macaulay local ring which possesses the canonical module $K_R$. Then $R$ is said to be an almost Gorenstein local ring, if there is an exact sequence $0 \to R \to K_R \to C \to ...

1
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0
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164
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### Local cohomology commuting with fiber

Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...

1
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0
answers

93
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### Upper bound for the minimum number of generators of the canonical module

Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$.
The question is that is there any upper bound for the ...

6
votes

1
answer

439
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### Matrix factorizations as a dg-category?

Matrix factorizations (in the graded case) give a triangulated category. I would imagine that there should be an underlying dg-category. Is there such a definition, and if so, where can I find it in ...

2
votes

2
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217
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### if $R$ is Noetherian local with a finite module of finite injective dimension and if "?" , then $R$ is "Gorenstein"

I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay.
Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...

4
votes

3
answers

515
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### canonical module can be identified with an ideal. how can one reach that ideal?

Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical
module, $K$. By Proposition 3.3.18 of Bruns_Herzog, $K$ can be identified with an ideal in $R$.
So we ...