Questions tagged [local-rings]
In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.
482 questions
1
vote
1
answer
45
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
3
votes
0
answers
54
views
Extending an automorphism of the residue field to an automorphism of the local ring.
Let $R$ be a commutative local ring with unity, and let $I$ be its maximal ideal.
Suppose there exists a ring automorphism $g: R/I \to R/I$.
Can we find a ring automorphism $f: R \to R$ such that the ...
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
1
answer
58
views
Lifting central units modulo Jacobson radical
Let $R$ be a finite unital ring (not commutative) and $J$ it's Jacobson radical. Assume that $R$ is local, that is, $R/J$ is a division ring and hence a finite field of characteristic $p$.
Since $Q = (...
- 3,030
2
votes
0
answers
74
views
The power series ring $R[[I]]$ for arbitrary index sets $I$
Throughout, $R$ is a commutative ring and $K$ is a field.
In my Algebra I class, today, we covered the power series ring $R[[X]]$. $K[[X]]$ was given as an example of a local ring.
Now, I know about ...
- 3,275
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
0
votes
0
answers
10
views
References on the claims of moduli spaces of additive compactifications by Hassett-Tschinkel
I am considering the classification problem of Artinian local $\mathbb{C}$-algebras, and notice the paper
Hassett, Brendan; Tschinkel, Yuri, Geometry of equivariant compactifications of (\mathbb{G}^...
- 1,479
2
votes
1
answer
47
views
Maximal ideal in the ring of formal power series in finitely many variables
Let $k$ be a field, $m \in \mathbb{N}$ and let $\{p_1,p_2,\ldots\}$ be the set of prime numbers in order. Consider the rings $P=k[[x_1,\ldots,x_m]]$ of formal power series in $x_1,\ldots,x_m$ ...
- 113
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
53
views
Local structure at ramification points of a morphism of smooth curves
Let $f:C \longrightarrow D$ be a morphism of smooth curves, simply branched and of degree $k$. We know that if $f$ is étale at $q \in C$ and $p=f(q) \in D$, then for $t$ a regular parameter in $\...
- 2,258
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 ...
0
votes
1
answer
38
views
Question about monogenic extension.
Let $A$ be a DVR with fraction field $K$ (let's say it's a global field for simplicity). Let $L/K$ be a finite (separable) extension, and let $B$ be the integral closure of $A$ in $L$. Must $B=A[\beta]...
- 817
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
1
answer
37
views
Valuation in the local ring of a curve
Let $k$ be an algebraically closed field and $A = k[X,Y]/(Y^2-X^3)$. Let $M$ be the ideal of $A$ generated by $X$, $Y$, and $A_M$ be the localization of $A$ at $M$.
Is $A_M$ a discrete valuation ring?...
- 1,165
0
votes
1
answer
59
views
Is $K[[X^2,X^3]]$ a local ring where $K$ is field? [closed]
I came a cross with this question "Is $K[[X^2,X^3]]$ a local ring where $K$ is field?" This is a ring of formal power series in which the terms are generated by $X^2$ and $X^3$ i.e. This ...
- 215
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
1
vote
1
answer
56
views
Does the uniformizer $\pi$ generate the extension $K/T$ where $T$ is the inertia field?
I'm trying to prove the equivalence of the statements
(1) $\sigma(x) \equiv x$ mod $\mathfrak{p}^{i+1}$ (that is, $\sigma \in V_i$, the $i^{th}$ ramification group) for all $x\in \mathcal{O}_{K,\...
- 539
0
votes
0
answers
14
views
$K^{ab}=K^{ur}M$, $M$ maximal, abelian, totally ramified
I read this property $K^{ab}=K^{ur}M$ where $M$ is a maximal totally
ramified abelian extension of $K$ (local field) as a corollary of the
following
Let $L/K$ abelian extension of a local field $K$. ...
- 329
0
votes
0
answers
47
views
Can Nakayama Lemma apply to complex
Let $R$ be a Noetherian commutative local ring with maximal ideal $\mathfrak{m}$. Consider a complex of finitly generated projective $R$-mod (so it is free)
$$0\rightarrow P_1\rightarrow P_2\...
- 1
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
0
votes
0
answers
44
views
How to compute $\operatorname{Length}(k[[x_1,\dots,x_n]]/I)$ for some ideal $I$
Let $k$ be an algebraically closed field and $\mathcal{O}_0=k[[x_1,\dots,x_n]]$ be the ring of formal power series, $I$ be an ideal of $\mathcal{O}_0$ such that $\operatorname{Spec}(\mathcal{O}_0/I)$ ...
- 977
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
0
votes
0
answers
45
views
Complete set of local orthogonal idempotents in $\mathbb{Z}_n$
I want to find a set of local orthogonal idempotents in $\mathbb{Z}_n$, i. e. such idempotents $e_1, \ldots e_n \in \mathbb{Z}_n$ that $1 = e_1 + \ldots + e_n$, $e_i e_j = e_j e_i = 0, (i \ne j)$, and ...
- 61
0
votes
0
answers
35
views
Proof that if $R \ne 0$ - commutative Artinian ring then $R \cong R_1 \times \ldots \times R_n$ [duplicate]
I'm trying to proof the following proposition. If $R \ne 0$ - commutative Artinian ring then $R \cong R_1 \times \ldots \times R_n$, where each $R_i$ is local Artian.
We known that fact, if $R$ is ...
- 61
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
22
views
Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$.
Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$.
The first thing I tried was to see that
$$\displaystyle\bigcap_{n = 1}^\...
- 3,830
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
2
votes
2
answers
110
views
A boolean ring which is local must be isomorphic to $\Bbb{F}_2$.
Recall: a Boolean ring is a (commutative) ring $R$ where $\forall x \in R: x^2=x$.
I don't really know how to proceed. I have tried some things like
If $x,y \ne 0$ in $R$ such that $x^2=x$ and $y^2=y$...
- 968
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 ...
- 1,855