All Questions
Tagged with local-rings commutative-algebra
313 questions
1
vote
1
answer
46
views
Understanding the Behavior of `isLicci(I)` in Macaulay2 for Squarefree Monomial Ideals
I am studying Licci squarefree monomial ideals, where the definition is typically given in the polynomial ring localized at the maximal ideal (Licci ideals can be defined more generally in a regular ...
- 359
2
votes
1
answer
103
views
Is this statement of the Nakayama Lemma incorrect?
In my lecture notes, I found the following proposition. I believe that it is incorrect.
Proposition (Nakayama for Local Rings). Let $(R, \mathfrak{m})$ be a commutative local ring with residue field $...
- 3,275
2
votes
1
answer
97
views
Is $\mathbb C[[x, y]]/(x^3+y^4)$ an integral domain?
The polynomial $x^3+y^4$ is irreducible in $\mathbb C[x, y]$, thus $\mathbb C[x, y]/(x^3+y^4)$ is an integral domain. My question is: Is $\mathbb C[[x, y]]/(x^3+y^4)$ also an integral domain?
Further ...
- 1,682
0
votes
1
answer
45
views
Two-dimensional affine quotient singularity modulo a linear element
Let $k$ be a field of characteristic $0$. Let $G$ be a finite subgroup of $GL(2, k)$ such that $\sigma- I_2$ does not have rank $1$ for every $\sigma\in G$. Consider the invariant subring $R := k[[x, ...
- 1,682
3
votes
1
answer
126
views
Socle of the maximal ideal and its square for a local ring with nonzero square of maximal ideal
Let $(R, \mathfrak m, k)$ be a Noetherian local ring such that $\mathfrak m^2\neq 0$. If $\text{Hom}_R(k, \mathfrak m^2)=0$, then is it true that $\text{Hom}_R(k, \mathfrak m)=0$?
My thoughts: Since $...
- 1,682
0
votes
1
answer
32
views
Module finite extension of one-dimensional local domain which is PID
Let $R$ be a Noetherian local domain of dimension $1$. Let $R \subseteq B$ be a module finite ring extension such that $B\subseteq Q(R)$. Then clearly, $B \subseteq \overline R$, where $\overline R$ ...
- 1,682
1
vote
1
answer
102
views
Embedding of modules and annihilation by maximal ideal
Let $(R,\mathfrak m, k)$ be a Noetherian local domain of dimension $1$. Let $x\in \mathfrak m$ be such that $\mathfrak m^{n+1}=x\mathfrak m^n$. Let $Y$ be a finitely generated torsion-free $R$-module ...
- 557
2
votes
0
answers
42
views
Localization of profinite integers
Let $p$ be a prime number, $\mathbb{Z}_{(p)}$ the localization of $\mathbb{Z}$ at $p$ and $\widehat{\mathbb{Z}} = \prod_{q\text{ prime}} \mathbb{Z}_q$ the profinite completion of the integers. Is the ...
- 3,030
1
vote
0
answers
46
views
Are two-periodic two-generated ideals of Gorenstein local domains self-dual?
Let $I$ be a non-principal ideal of a Gorenstein local domain $R$ such that there is an exact sequence $0\to I \to R^{\oplus 2}\to R^{\oplus 2}\to I\to 0$.
Then, must it be true that $I\cong \text{Hom}...
- 1,682
0
votes
1
answer
52
views
Indecomposability of an object, in the bounded derived category of finitely generated modules, whose derived self-Hom is isomorphic to the ring
Let $R$ be a commutative Noetherian local ring. Let $M\in \mathcal D^b(\text{mod } R)$ be such that $\mathbf R \text {Hom}_R(M, M)\cong R$. Then, is $M$ indecomposable in $\mathcal D^b(\text{mod } R)$...
- 4,476
4
votes
1
answer
147
views
Finite map of local rings satisfying certain properties is surjective
$\newcommand{\iM}{\mathfrak{m}}\newcommand{\iN}{\mathfrak{n}}$I'm working on the following exercise and am quite stuck.
Let $(A,\iM)$ and $(B,\iN)$ be local rings, and let $f:A\to B$ be a map of local ...
0
votes
1
answer
41
views
Length of quotient ring by power of maximal ideal
Let $(R, \mathfrak{m})$ be a local ring such that $\mathfrak{m}$ is generated by $k$ elements. Then for every $n$, what is the length of the module $R/\mathfrak{m}^n$?
We know we have an exact ...
0
votes
1
answer
96
views
Is a localization of a local ring local?
I'm trying to answer the following question: " If $R$ is a local ring and $S$ a multiplicative set, is $S^{-1}R$ a local ring?"
I found out in this other question that a localization is ...
2
votes
1
answer
84
views
Is there a 1-periodic module M over a local non-Cohen Macaulay ring?
As the title states, I am looking for an example of a module $M$ (non-zero and non-free) over a non-Cohen-Macaulay local Noetherian ring $(R,\mathfrak m, k)$ where $M$ is periodic of period one. There ...
3
votes
0
answers
162
views
Completion of primary ideal is primary
I have the following question.
Suppose $(R, \mathfrak{m})$ is a local Noetherian ring, $\mathfrak{a}$ is $\mathfrak{p}$-primary ideal. Is it true that $\hat{\mathfrak{a}}$ is a primary ideal in $\hat{...
- 105
0
votes
0
answers
47
views
Dimension of graded rings
Let $A=\oplus_{i \in N \cup \{0\}} A_i$ be a positively graded ring of dimension $d$ with $A_0=k$ and $k$ is a field. If $B$ is a Noetherian graded subring of $A$. Can we say dimension of $B$ cannot ...
- 101
0
votes
1
answer
32
views
Minimal generator systems of a finitely generated
Let $A$ a local commutative ring with unity and $M$ an $A$-module. If $M$ is finitely generated, its minimal generator systems have the same cardinal.
I am trying to prove this claim in order to use ...
- 496
1
vote
1
answer
72
views
Cohen--Macaulay rings $R$ of dimension more than $1$ such that $R/\mathfrak p$ is also Cohen--Macaulay for every minimal prime ideal $\mathfrak p$
Let $R$ be a Cohen--Macaulay local ring. If $\dim R=1$, then $R/\mathfrak p$ is also a Cohen--Macaulay ring for every minimal prime ideal $\mathfrak p$ of $R$. It is known that for $\dim R\ge 2$, this ...
- 435
1
vote
0
answers
56
views
Is there a "minimal free resolution" for non-finitely generated modules?
Let $(R,\mathfrak m)$ be a commutative local ring and $M$ be an $R$-module such that $\mathfrak m^n M=0$ for some $n>0$. Then, does there exist a projective (necessarily free by Kaplansky's theorem)...
- 341
0
votes
0
answers
62
views
Using Nakayama's lemma in non-local ring
Let $R$ be a Noetherian integral domain of dimension one and $\mathfrak{m}$ an ideal such that $\text{dim }\mathfrak{m}/\mathfrak{m}^2=1$ as an $R/\mathfrak{m}$-vector space. The localization of $R$ ...
- 323
1
vote
1
answer
36
views
Uniformizers in one-dimensional local rings
Suppose that $R$ is a Noetherian ring with unique maximal ideal $\mathfrak{m}$. Further suppose that $\mathfrak{m}/\mathfrak{m}^2$ is a one-dimensional $R/\mathfrak{m}$-vector space. By an application ...
- 323
2
votes
1
answer
95
views
When is the initial form of a principal ideal generated by the initial form of the original ideal's generator?
$
\DeclareMathOperator{\init}{in}
\DeclareMathOperator{\gr}{gr}
\newcommand{\calO}{\mathcal{O}}
$Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\gr_{\mathfrak{m}}(R)=\bigoplus_{i=0}^\infty\...
- 265
0
votes
1
answer
55
views
Possible inequality of krull dimension of local injection of Noetherian local domains
If $(A, \mathfrak{m}) \hookrightarrow (B, \mathfrak{n})$ is a local injection of Noetherian local domains, do we necessarily have $\dim B \geq \dim A$?
- 567
1
vote
1
answer
214
views
Finiteness of the intersection multiplicity of plane algebraic curves
Hello guys i am trying to solve excerise 2.7 page 14 from Gathmann notes https://agag-gathmann.math.rptu.de/class/curves-2023/curves-2023-c2.pdf
Definition
About (a) : Stuck here.Not sure how to ...
- 31
1
vote
1
answer
40
views
How to compute a minimal spanning set and the minimal spanning number of $\Bbb C[x,y]_{(x-1,y-1)}/(x^3-y^2)$?
Let $A=\frac{\mathbb C[X,Y]}{(X^3-Y^2)}$. I am asked to show that $\mathbf m=(\overline X-1,\overline Y-1)$ is a maximal ideal of $A$ which I have shown successfully. Now I am asked to compute the $\...
- 3,801
3
votes
0
answers
99
views
Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?
Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
- 435
1
vote
0
answers
46
views
How to understand a situation where one can use Nakayama's lemma even when the situation is not Tailor-made.
In commutative algebra we have the following version of Nakayama's Lemma(also calle NAK lemma):
NAK Lemma:
Let $R$ be a local ring and $\mathbf m$ be the unique maximal ideal of $R$.Let $M$ be a ...
- 3,801
0
votes
1
answer
78
views
Algebraic properties of the ring of analytic functions on the complex plane
Let $r, R>0$ and $D_{R}=\{z\in\mathbb{C}: \|z\|<R\}$ and $D_{r,R}=\{z\in\mathbb{C}: r<\|z\|<R\}$. Consider the following
$\mathcal{R}_{1}:=\mathcal{O}_{R}=\{f:D_{R}\to\mathbb{C}: f\hspace{...
- 675
1
vote
1
answer
56
views
Dominating rings: $\mathfrak{m}_A=A\cap \mathfrak{m}_B\Longleftrightarrow \mathfrak{m}_A\subset \mathfrak{m}_B$
Exercise $27$ from Atiyah and MacDonald states that if $A,B$ are two local rings, then $B$ is said to dominate $A$ iff $A$ is a subring of $B$ and the maximal ideal $\mathfrak{m}_A$ of $A$ is ...
- 2,096
-4
votes
1
answer
94
views
Quotients of local rings [closed]
I have the following commutative algebra question. It came up at the time of reading local rings of schemes and their tangent spaces.
Let $R$ be a local ring with unique maximal ideal $m$. Then is it ...
- 275
0
votes
0
answers
81
views
Free module over local ring $R$. [duplicate]
People often say that a module $M$ over a not necessarily neotherian local ring $R$ being projective is flat, and also free. However, some refer to the finitely-generatedness of $M$, i.e. $M$ being ...
- 893
3
votes
1
answer
109
views
Finitely generated torsion-free indecomposable modules over one-dimensional complete local domains are isomorphic to ideal?
Let $R$ be a complete local domain of dimension $1$. Let $M$ be a finitely generated torsion-free indecomposable $R$-module. Then, must $M$ be isomorphic to an ideal of $R$?
Also clearly, $R$ embeds ...
- 1,682
1
vote
1
answer
86
views
Show that $A[T]/(P(T))$ is a local flat A-algebra where $(A,\mathfrak{m},k)$ is a local ring with $k[T]/(\tilde{P}(T))$ is a simple extension of $k$
Let $(A,\mathfrak{m},k)$ be a (Noetherian) local ring. Let $k[T]/(\tilde{P}(T))$ be a simple field extension of $k$ with $\tilde{P}(T)$ monic and irreducible (and separable). Lift $\tilde{P}(T)$ to ...
- 1,865
1
vote
0
answers
40
views
On the finiteness of certain Zariski closed subsets of the prime spectrum of commutative Noetherian local rings
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S,T$ be Zariski closed subsets of $\text{Spec}(R)$ such that if $\mathfrak p\in S, \mathfrak q \in \text{Spec}(R)$ and $\mathfrak p \subsetneq \...
- 1,682
5
votes
0
answers
86
views
On idempotents and local property of $\Lambda \otimes_R R_{\mathfrak p}$ where $\Lambda$ is module finite algebra over commutative local ring $R$
Let $R$ be a commutative Noetherian local ring. Let $\Lambda$ be a module finite associative $R$-algebra. Let $\mathfrak p$ be a prime ideal of $R$. I have the following two questions:
(1) If $\...
- 1,682
1
vote
0
answers
89
views
Proving two modules are free based on their direct sum [duplicate]
I have given two modules $M$ and $N$ over a local ring $R$. I also know that $M \oplus N \cong R^n$ for some $n\in \mathbb{N}$. I then have to prove that both $M$ and $N$ are free modules.
Since $M \...
- 139
1
vote
1
answer
72
views
How to express elements of a DVR as power series of the uniformizer?
Let $R$ be a DVR with a fixed uniformizer $t$. My main considering example is the local ring of an algebraic curve over an algebraically closed field $k$, so let's assume $R$ is a $k$-algebra and has ...
- 914
0
votes
1
answer
71
views
Is a flat extension of DVRs always finite?
Is it true that if $S/\mathbb{Z}_p$ is a flat extension of discrete valuation rings then $S$ is finite over $\mathbb{Z}_p$?
- 527
1
vote
0
answers
82
views
Module over a commutative local ring has nonzero homomorphism
The question is from a Persian text book:
Let $R$ be a commutative local ring with $1$, and $\mathfrak m$ its maximal ideal. If $M$ is a non-zero module over $R$ prove that $\text{Hom}_R(M,R/\mathfrak ...
- 136
3
votes
1
answer
92
views
Serre conditions and local isomorphisms
I encountered the following in the introduction of the paper "Duality for Koszul Homology over Gorenstein Rings":
I assume it's easy, but why is this fact true? I played with modding out by ...
- 1,460
-3
votes
1
answer
107
views
How can I show that if $x$ is a nonunit then $1-x$ is a unit? [duplicate]
Let $R$ be a commutative local ring. I want to show that then $x\in R$ is not a unit implies $1-x$ is a unit.
My idea was the following:
Since $R$ is a local ring we can chose $\mathfrak{m}$ to be ...
- 297
1
vote
1
answer
70
views
On pushforward functor and freeness of modules
Let $R$ be a commutative Noetherian ring. Given an $R$-module $M$ and a ring homomorphism $\phi:R\to R$, let $\phi^* M$ be the $R$-module whose underlying abelian group is the same as $M$ but the $R$-...
- 435
0
votes
0
answers
111
views
Betti numbers of associated graded ring
Let $(R,\mathfrak{m})$ be a (not necessarily commutative) local ring with commutative residue field $k$. Define it's betti numbers as
$$\beta_i(R):=\dim_k\mathrm{Tor}^i_R(k,k).$$
The associated graded ...
- 1,270
3
votes
0
answers
114
views
Is the ring of formal power series a localization of some non local ring at prime ideals?
Consider $F = K[[x]],$ the formal series over field $K.$ We know that $K$ is a local ring with maximal ideal $(x).$ Does there exist a non local ring $R$ and prime ideal $p$ such that $R_p = K[[x]]?$ ...
- 793
1
vote
1
answer
55
views
Do we have $\bigcap_{1\leq i\leq r}(\mathfrak{m}^n+\mathfrak{p}_i)=\mathfrak{m}^n$ in a local ring, where $\mathfrak{p_i}$ are the minimal primes?
$\newcommand{\cO}{\mathcal{O}}\newcommand{\fm}{\mathfrak{m}}\newcommand{\fp}{\mathfrak{p}}$
Let $(\cO,\fm)$ be a Noetherian reduced local ring, essentially of finite type over an algebraically closed ...
- 2,839
1
vote
0
answers
89
views
Finiteness of the homology of the Koszul complex
Let $A$ be a local noetherian commutative ring, let $x_1, \dots, x_r$ be elements of the maximal ideal of $A$, $I$ the ideal that they generate, $M$ a $A$-module of finite type such that $M/IM$ is of ...
1
vote
0
answers
73
views
A sort of "minimal presentation " for a local ring essentially of finite type over a field
Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
- 341
1
vote
0
answers
78
views
$1$-dimensional reduced local ring $R$ such that $(R:_{Q(R)} \overline R)$ is non-zero and consists of zero-divisors
This question is motivated by If $(R:_{Q(R) } S)$ is non-zero, then does $(R:_{Q(R) } S)$ contain a non-zero-divisor? (and a now deleted comment on it).
Namely: Does there exist a one-dimensional ...
- 65
0
votes
1
answer
78
views
Quotient of the polynomial ring over a local noetherian ring is finitely generated?
I am going through these lecture notes, and I am confused about the proof of Corollary 10.10 there.
Corollary 10.10. Let $A$ be a local noetherian domain with maximal ideal $\mathfrak{p}$, let $g \in ...
- 179
2
votes
1
answer
106
views
If $M\cong N \oplus k^{\oplus a}$, where $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$, then $k$ is not a direct summand of $N$?
Let $M$ be a finitely generated module over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $M\cong N \oplus k^{\oplus a}$, where $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$...
- 105