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Questions tagged [cohen-macaulay-rings]

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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 ...
klerk's user avatar
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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 ...
Snake Eyes's user avatar
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\...
Alex's user avatar
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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 ...
Snake Eyes's user avatar
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 $R/\mathfrak{p}$ ...
delusional.existence's user avatar
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 ...
uno's user avatar
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2 votes
0 answers
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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_{\...
walkar's user avatar
  • 253
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 ...
uno's user avatar
  • 280
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 $R$ ...
Snake Eyes's user avatar
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 $...
Damian Rössler's user avatar
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 $...
Snake Eyes's user avatar
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}...
strat's user avatar
  • 349
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)=\...
Serge the Toaster's user avatar
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 ...
Mare's user avatar
  • 26.2k
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 ...
Alex's user avatar
  • 365
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 $\text{Tor}_n^R(R/I, R/J)$ ...
Alex's user avatar
  • 365
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#...
Alex's user avatar
  • 365
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 \...
Serge the Toaster's user avatar
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 ...
user782932's user avatar
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 ...
pink floyd's user avatar
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 ...
Misha Verbitsky's user avatar
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 $\...
user45397's user avatar
  • 2,293
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 ...
Danimenru's user avatar
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 ...
V.S.'s user avatar
  • 111
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 ...
Tom Gannon's user avatar
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 ...
Louis 's user avatar
  • 279
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 ...
Joël's user avatar
  • 25.8k
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), ...
pink floyd's user avatar
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&...
strat's user avatar
  • 349
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. ...
user237522's user avatar
  • 2,783
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 ...
user237522's user avatar
  • 2,783
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 ...
user521337's user avatar
  • 1,199
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-...
Alessio's user avatar
  • 401
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 $M$...
AGBeginner's user avatar
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 ...
Jef's user avatar
  • 889
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 ...
One More Question's user avatar
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....
Nick Addington's user avatar
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 $(...
user160167's user avatar
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 ...
Louis 's user avatar
  • 279
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 ...
sdey's user avatar
  • 642
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 ...
user267839's user avatar
  • 6,044
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 ...
user45397's user avatar
  • 2,293
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})...
Horstenson's user avatar
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 ...
Paolo1994's user avatar
  • 113
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 ...
User43029's user avatar
  • 596
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 ?
user521337's user avatar
  • 1,199
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 ...
Ale's user avatar
  • 71
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 ...
Lisa S.'s user avatar
  • 2,633
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 $\mathfrak{p}$, $\...
Homa81's user avatar
  • 191
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 ...
LeLoupSolitaire's user avatar