# Questions tagged [cohen-macaulay-rings]

The cohen-macaulay-rings tag has no usage guidance.

89
questions

3
votes

1
answer

269
views

### Is the subscheme parametrizing the k-th degeneracy loci Cohen-Macaulay?

Let $X$ be a Cohen-Macaulay projective variety (say over an algebraically closed field of characteristic 0), and $E,F$ two vector bundles such that $E^*\otimes F$ is globally generated. Consider the ...

2
votes

1
answer

91
views

### A perfect complex over a local Cohen--Macaulay ring whose canonical dual is concentrated in a single degree

Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a ...

2
votes

0
answers

71
views

### From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences

Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...

3
votes

1
answer

224
views

### Vanishing of $\operatorname{Ext}_R^{1}(M,R)$ when $R$ is a Gorenstein local ring of dimension $1$ and $M$ is not finitely generated

Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...

1
vote

0
answers

99
views

### Quotient of a polynomial ring with a prime ideal is Cohen$-$Macaulay

[Bruns-Herzog, Exercise 2.1.17] Let $k$ be a field and $R = k[x_1, . . . , x_n]$. Suppose $\mathfrak{p} \subset R$ is a prime ideal, $ht\mathfrak{p} \in \{0, 1, n − 1, n\}$. Show that $R/\mathfrak{p}$ ...

0
votes

1
answer

130
views

### Is the integral closure of a henselian local domain of dimension $1$ again local?

Let $(R,\mathfrak m)$ be a local domain of dimension $1$. Let $\overline R$ be the integral closure of $R$ in the field of fractions $Q(R)$.
If $R$ is henselian, then is $\overline R$ also a local ...

2
votes

0
answers

114
views

### An alternative proof that Buchsbaum rings are generalized Cohen-Macaulay

Let $(R,\mathfrak{m})$ be a Noetherian local ring. $R$ is said to be Buchsbaum if, for each ideal $\mathfrak{q}$ generated by a full system of parameters, the number $\lambda_R(R/\mathfrak{q})-e_{\...

2
votes

1
answer

141
views

### Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$

Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...

2
votes

2
answers

358
views

### Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?

Let $k$ be a field of characteristic $0$. Let $R$ be a Noetherian local normal domain containing $k$. Also assume that $R$ is the homomorphic image of a Gorenstein ring of finite dimension, hence $R$ ...

4
votes

1
answer

294
views

### Example of a certain type of Cohen-Macaulay ring

Let $k$ be a perfect field. I am looking for a $k$-algebra $R$ with the following properties.
$R$ is of finite type over $k$ and is a domain;
for all ${\mathfrak p}\in{\rm Spec}(R)$, the local ring $...

1
vote

0
answers

98
views

### On Serre's condition and singular locus of determinantal rings

Let $R$ be a Commutative Noetherian ring. Let $\mathbf X:=[X_{ij}]_{1\le i \le r, 1 \le s \le t}$ be a matrix of indeterminates. Let $t>1$ be an integer, and $I_t(\mathbf X)$ denote the ideal in $...

2
votes

0
answers

95
views

### When does a local Cohen–Macaulay ring admit a non-zero finitely generated maximal Cohen–Macaulay module of finite injective dimension?

Let $(R,\mathfrak m)$ be a local Cohen–Macaulay ring. Then, it is well- nown that there exists a non-zero finitely generated $R$-module of finite injective dimension; for instance $\operatorname{Hom}...

0
votes

1
answer

264
views

### Depth of almost complete intersection rings

Let $R$ be a regular local ring and let $I \subset R$ be an almost complete intersection ideal, that is, $\mu(I)=\text{ht}(I)+1$ where $\mu(I)$ is the number of minimal generators of $I$ and $ht(I)=\...

2
votes

0
answers

177
views

### Commutative local rings which satisfy Krull-Remak-Schmidt

Question 1: Can the class of local (always noetherian and commutative) rings be classified for which the Krull-Remak-Schmidt theorem (KRS) holds for finitely generated modules? They contain for ...

6
votes

1
answer

301
views

### When can we choose non-zero-divisor $x\in \mathfrak m$ in a reduced local ring $(R,\mathfrak m)$ such that $R/xR$ is also reduced?

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay reduced ring of dimension at least $2$. Then, can we find a non-zero-divisor $x\in \mathfrak m$ such that $R/xR$ is again a reduced ring?
If needed, I ...

6
votes

2
answers

261
views

### If Serre's intersection multiplicity $\chi(R/I, R/J)$ equals $\operatorname{length}_R (R/(I+J))$, then are $R/I, R/J$ Cohen-Macaulay?

Let $(R,\mathfrak m)$ be a regular local ring. Let $I,J$ be proper ideals of $R$ such that $R/(I+J)$ has finite length i.e. $\sqrt{I+J}=\mathfrak m.$ Since $I+J$ annihilates $\text{Tor}_n^R(R/I, R/J)$ ...

2
votes

1
answer

304
views

### When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at prime ideals?

For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#...

1
vote

1
answer

140
views

### Cohen-Macaulay quotient ring and symbolic power

Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{a} \subset R$ be an ideal. Let
$$ \mathfrak{b} = \bigcap \{R \cap \mathfrak{a} \cdot R_\mathfrak{p} \text{ } \colon \mathfrak{p} \in \...

2
votes

0
answers

161
views

### 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 ...

4
votes

1
answer

210
views

### 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 ...

8
votes

2
answers

792
views

### 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

127
views

### 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
answers

550
views

### 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
vote

0
answers

121
views

### 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 ...

10
votes

1
answer

567
views

### 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

422
views

### 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
answers

312
views

### 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

556
views

### 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

210
views

### 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
vote

0
answers

111
views

### $A \to B$ 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

224
views

### 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

188
views

### 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

251
views

### 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
vote

1
answer

287
views

### 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 $M$...

12
votes

1
answer

396
views

### 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

188
views

### 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

261
views

### 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

127
views

### 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

103
views

### 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

264
views

### 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
votes

0
answers

164
views

### 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

195
views

### 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

186
views

### 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

630
views

### 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

682
views

### 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

265
views

### 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
votes

0
answers

209
views

### 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

681
views

### 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

186
views

### 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

191
views

### 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 ...