All Questions
Tagged with ac.commutative-algebra cohen-macaulay-rings
21 questions with no upvoted or accepted answers
7
votes
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289
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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 ...
- 2,851
5
votes
0
answers
132
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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 $(...
- 51
5
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0
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106
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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 ...
- 279
5
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0
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202
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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 ...
- 2,323
5
votes
0
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248
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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?
4
votes
0
answers
212
views
A doubt from the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring"
I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff ...
- 1,713
3
votes
0
answers
351
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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 ...
- 26k
3
votes
0
answers
199
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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})...
- 373
3
votes
0
answers
293
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Simple proof of that $k[X]^G$ Cohen-Macaualy ($G$ finite)?
Let $X$ be a (EDIT: non-singular, or even $\mathbf A^n$) algebraic variety over a field $k$ (alg. closed). Suppose $G$ is a finite group acting on $X$, $|G|\neq 0$ in $k$. Then $k[X]^G$ is Cohen-...
- 31
2
votes
0
answers
73
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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\...
- 480
2
votes
0
answers
123
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_{\...
- 253
2
votes
0
answers
109
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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}...
- 361
2
votes
0
answers
196
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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 ...
- 26.5k
2
votes
1
answer
388
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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#...
- 480
2
votes
0
answers
172
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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 ...
- 131
1
vote
0
answers
97
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Cohen-Macaulayness of the homogeneous coordinate ring of projective monomial curves
Let $A = \{a_0, a_1, \ldots, a_{n-1}\} \subset \mathbb{N}$ be a set of non-negative integers where we assume that $a_0 < \cdots < a_{n-1}$ and set $d := a_{n-1}$. For every $s \in \mathbb{N}$, ...
- 21
1
vote
0
answers
131
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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 $R/\mathfrak{p}$ ...
1
vote
0
answers
108
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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 $...
- 413
1
vote
0
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149
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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 ...
- 111
1
vote
0
answers
172
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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 $A$-...
- 5,562
1
vote
0
answers
106
views
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 ...