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.

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Ring of total fractions of the strict henselization of a non-normal local ring

Let $A$ be an integral Noetherian local ring of dimension 1 which is not normal and with residue field a finite field $\mathbb{F}_q$ (so typically some local ring of a singular curve on a finite field)...
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Subset of $\operatorname{End}_A(M)$ faithful implies $M$ is faithful.

Let $A$ be a complete intersection ring, $M$ be a finite $A$-module of positive depth (over the maximal ideal), and $B$ be the image of $A$ in $\operatorname{End}_A(M)$. It is easy to show that $B$ ...
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Reduced noetherian local ring of depth zero is artinian?

It is well-known that a local ring $A$ with maximal ideal $\mathfrak{m}$ of depth zero is not necessarily artinian (e.g. $k[x,y]/(xy, x^2)$ localised at the origin), but what if we further require ...
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4 votes
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Codimension inequality of prime ideal in a regular local ring

The following is an exercise (#6) of Eisenbud's commutative algebra, chapter 10: Exercise. We mentioned that if $P$ is a prime ideal in a regular local ring $R$ and if $R\to S$ is a map of local rings,...
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2 votes
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Ideal determined by completions

Let $\mathcal{O}$ be an order in a number field $K$. Question: Is specifying an invertible ideal $I$ of $\mathcal{O}$ equivalent to specifying invertible ideals $I_p$ of $\mathbb{Z}_p \otimes_{\mathbb{...
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Why $\mathfrak{p}A_\mathfrak{p} = 0$, where $A_{\mathfrak p}$ is the localization at the kernel $\mathfrak p$ of a surjective ring homomorphism.

Let $A$ be a commutative, Noetherian, local ring, $\mathfrak{O}$ a discrete valuation ring and $\lambda : A \rightarrow \mathfrak{O}$ be an epimorphism. Let $\mathfrak{p}=\ker(\lambda)$, and consider ...
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Is every parameter outside of the union of minimal primes?

Proposition 1: Let $R$ be a commutative, Noetherian ring and $\ \mathfrak{p} \in \operatorname{Spec}(R)$. If $\operatorname{ht}(\mathfrak{p})=h$, then there exist $y_1, \ldots, y_h \in R$ such that $\ ...
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Finitely generated modules over a noetherian local ring whose completions are isomorphic

This is part of an exercise in Eisenbud's commutative algebra, chapter 7, no. 5. Suppose $M$ and $N$ are finitely generated modules over a Noetherian local ring $R$ whose completions $\hat{M}$ and $\...
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2 votes
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Zerodivisors on quotient module [closed]

Let $R$ be a Noetherian local ring and $M$ be a finitely generated $R$-module. Suppose that two ideals $I$ and $(x)$ are consisting of zerodiviors on $M$. Given a nonzerodivisor $a\in (I,x)$ on $M$, ...
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On $k$-vector space dimension of $R/(k+I)$ for a Noetherian local ring $(R,\mathfrak m,k)$ containing $k$

Let $(R,\mathfrak m,k)$ be a Noetherian local ring. Assume $R$ contains $k$. Let $I$ be an ideal of $R$ such that $\ell_R(R/I)<\infty$. Consider $k+I:=\{a+b: a\in k, b\in I\}$. Then, $k+I$ is a sub-...
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Minimal homogeneous generating sets in graded rings

Let $R$ be a Noetherian $\mathbb{N}$-graded ring where $R_0=K$ is an infinite field. I explicitly do not want to assume that $R$ is standard graded (i.e. $R\neq K[R_1]$). If $I$ is a homogeneous ideal ...
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2 votes
1 answer
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Local rings of $V(y^2-x^3)$

I want to find the local rings of $V(y^2-x^3)$, and establish if it's isomorphic to $K[x]_{(x)}$, or maybe some other ring which I don't know. We take $p=(t^2,t^3) \in V$ and we want to find $O_{(t^2,...
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Modules over a DVR can be considered as modules over its congruence algebra?

Let $A$ be a commutative, Noetherian, local ring, $O$ a discrete valuation ring and $\lambda : A \rightarrow O$ be an epimorphism. Let $p:=\ker(\lambda), \Phi_A:= p/p^2, I:=Ann[p], \Psi_A:=O/\lambda(I)...
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Condition for conormal module of commutative, Noetherian, local ring to have finite length

Let $A$ be a commutative, Noetherian, local ring, $O$ a discrete valuation ring and $\lambda : A \rightarrow O$ be an epimorphism. Let $p=\ker(\lambda)$, and consider the conormal $A$-module $p/p^2$. ...
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Minimal number of generators and base change

For every finitely generated module $M$ over a Noetherian local ring $(R,\mathfrak m,k)$, every minimal generating set of $M$ has the same cardinality and that is given by $\dim_k(M\otimes_R k)=\dim_k(...
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3 votes
1 answer
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Two Properties of Perfect Rings

I came across Kunz's theorem about the characterization of regular rings in characteristic $p$. In the paper that I am reading, the author uses perfect rings to prove this result. Perfect rings $R$ ...
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2 votes
1 answer
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Are uniformizers of DVR's unique?

I just started learing about discrete valuation rings, so I don't know a lot of examples and I can't find any counterexamples. So consider a DVR $(R,v)$ with valuation group $\Gamma_v \neq \{0\}$. A ...
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7 votes
1 answer
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Why are local rings called local?

I gather that rings of germs of functions at a point $p$ on a manifold/variety/etc. are local with the maximal ideal containing exactly the germs of functions which vanish at $p$. So in some sense, ...
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1 answer
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Question about a proof involving local rings: $R$ has exactly $3$ ideals, show that if $a,b\in I$ then $ab=0$

I have a question regarding the following thread: Commutative unitary ring with exactly three ideals. I believe I've put the pieces together, but I am, for whatever reason, feeling uncomfortable. So, ...
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3 votes
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What does finding a "free local ring" have to do with finding the spectrum of a ring?

In Tierney's 1976 paper On the Spectrum of a Ringed Topos (which you can find here) at the top of section 2 we read Let $A$ be a commutative ring in [a topos] $\mathbf{E}$. We look at the problem of ...
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2 answers
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Support of module and faithfully flat base change

Let $R \subseteq S$ be a faithfully flat extension of Noetherian local rings. Let $M$ be a finitely generated $R$-module such that $\operatorname{Supp}_R(M)=\operatorname{Spec}(R)$. Then, is it true ...
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11 votes
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For local ring $R$, does funcotor $\operatorname{Hom( Spec}R, X)$ characterize scheme $X$?

Let $\bf{Sch, Sets, Ring}$ be a category of schemes, sets, commutative rings. By Yoneda's lemma, scheme $X$ is characterized by contravariant functor $$\operatorname{Hom}(*, X): \bf{Sch}^{op}\to Sets$$...
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Computing residue fields of affine schemes

I have taken a course on schemes, so I am familiar with the basic definitions, but I'm very rusty and I've forgotten how to do this (if I ever knew). Basically, I want to compute the residue fields of ...
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0 votes
1 answer
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Can a discrete valuation ring be finitely generated over a field?

In my homework of schemes, the professor proposed the following exercise: "Let $ X $ be a scheme of finite type over a field $ k $ and $ f \in \mathcal{O}_X (X) $ a global section. Show that $ f $...
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1 answer
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Regular local rings of Krull dimension $1$ are integral domains

I am reading a proof of the fact that any regular local ring $R$ of Krull dimension $1$ is an integral domain. It was previously shown that the maximal ideal $\mathfrak m$ of $R$ is generated by some $...
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Family of generators of a module over a local ring [duplicate]

I have a local ring $(R,\mathfrak m)$ and a module $M$ which is free of finite rank ($M\simeq R^n$, for some $n\in\mathbb N$). Given a family of generators $v_1,\cdots,v_m$ I know that I can extract a ...
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1 vote
1 answer
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If $\operatorname{Idem}B\rightarrow \operatorname{Idem}(B/mB)$ is surjective then $B$ is a product of local rings

Let $A$ be a local ring with maximal ideal $m$ and $B$ a finite $A$-algebra (by finite I mean that $B$ is a finitely generated $A$-module). If we denote by $\operatorname{idem}B$ (respectively $\...
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7 votes
1 answer
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Why does a finite module over a Noetherian local ring supported only at the maximal ideal have the residue field as a submodule and a quotient?

I am reading the book “Fourier-Mulkai transforms in algebraic geometry” by Daniel Huybrechts. In the proof of Lemma 4.5, in page 92, it is written that if $M$ is a finite module over a local ...
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1 vote
1 answer
64 views

Elementary proof of injection from $\operatorname{Hom}(R, k[\varepsilon])$ to $\operatorname{Hom}_k(\mathfrak{m}_R /\mathfrak{m}_R^2, k)$

Let $k$ be a finite field, let $\mathbf{CR}$ be the category of complete local Noetherian rings, and let $\mathbf{CR}_{/k}$ be the over category. Let $\mathcal{C}$ be the full subcategory of $\mathbf{...
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4 votes
1 answer
157 views

A proof that if A is a local ring, then the set of non-units is an ideal, without using Zorn's Lemma?

I need to prove that if $A$ is a local ring, then the set of non-units form an ideal. To do this it is suggested that I should use Zorn's lemma, but I seem to have found a way to do without. In ...
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0 votes
1 answer
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Understanding the structure of $\mathbb{Z}[[x]]/(x-x^2)$

I'm currently trying to understand the structure of quotients of power series rings, and found a particular example I'm confused about. Let $f = x-x^2$ be a polynomial in $\mathbb{Z}[[x]]$, and ...
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2 votes
2 answers
152 views

Is there any isomorphism between these two local rings?

Are the following two local rings isomorphic for any prime number $p$ and field with p elements, $\mathbb{Z}_p$ : $$\frac{\mathbb{Z}_p[X,Y]}{(X^3,XY,Y^2)}\ \text{ and } \frac{\mathbb{Z}_p[X,Y]}{(X^3,X^...
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3 votes
0 answers
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On the natural map $\Phi_M: M\otimes_RM^*\rightarrow\text{Hom}_R(M^*,M^*),\ x\otimes y\mapsto \left\{f\mapsto f(x)y\right\} $

Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\text{Hom}_R(\_,R)$. There is a natural map \begin{align} \Phi_M: M \otimes_R M^* & \...
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  • 129
0 votes
1 answer
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Does the torsion submodule and the $0$-th local Cohomology module coincide over local Cohen-Macaulay ring? [closed]

Let $M$ be a finitely generated module over a local Cohen-Macaulay ring $(R,\mathfrak m)$. If $x\in M$ is annihilated by a non-zero-divisor $r\in \mathfrak m$ , then is it true that $\mathfrak m^n x=0$...
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2 votes
1 answer
113 views

If a module is second syzygy and has no free summand , then it can be defined via a minimal resolution?

Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Assume $M$ has no free summand, and that there exists an exact sequence $0\to M \to R^{\oplus a}\xrightarrow{f} R^...
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2 votes
2 answers
92 views

If $\text{Tor}^R_1(M,R/xR)=0$, then is $x$ necessarily $M$-regular?

Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Let $x\in \mathfrak m$ be such that $\text{Tor}^R_1(M,R/xR)=0.$ Then, is it true that $x$ is $M$-regular? I can ...
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1 vote
1 answer
56 views

When an element of the total ring of fractions of a noetherian local ring $R$ is integral over $R$?

Let be $(R,\mathfrak{m},k)$ a Noetherian local ring. Denote by $Q$ the total ring of fractions of $R$. Let $I$ be an ideal of $R$ and $r \in I$ a $R$-regular element. I have the following question: ...
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4 votes
0 answers
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Question about socle in Huneke, Sega and Vraciu's paper.

I am reading Huneke, Sega, and Vraciu's paper Vanishing of Ext and Tor over Cohen-Macaulay local rings. There is a statement in their paper which I can't understand. Let $(R,\mathfrak{m},k)$ be an ...
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1 vote
1 answer
95 views

Questions regarding solution to showing: $M\otimes_A N = 0 \implies M = 0$ or $N = 0$.

I'm trying to understand the proposed solution of the following exercise: Let $A$ be a local ring and $M,N$ finitely generated $A$-modules. Show that $$M\otimes_A N=0$$ already implies $M = 0$ or $N=...
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2 votes
1 answer
29 views

A basic question on commutative finite local rings

Let $R$ be a commutative finite local ring of order $p^n$ ($p$ is a prime and $1\in R$). I'm struggling with the following two basic questions: (a) Is it true that $x^n=0$ for every non-unit $x\in R$ ?...
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0 votes
1 answer
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surjection of complete noetherian local rings -reference help

I am looking Lemma 1.1: $\widehat{C}$ is the category of complete local noetherian rings. Is there a reference to the last two lines of the above argument ? i.e. : associated graded rings being ...
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-1 votes
1 answer
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Noetherian local ring of depth $1$ whose localization at some prime ideal has higher depth? [closed]

Let $(R,\mathfrak m)$ be a Noetherian local ring of depth $1$. Then, is it possible that depth$(R_P)\ge 2$ for some prime ideal $P$ of $R$? Of course such an example has to be non-Cohen-Macaulay, i.e. ...
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5 votes
0 answers
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derivations of the ring of germs of $C^{\infty}$ functions

Let $\mathcal{O}_{\mathbb{R},0}$ be the ring of germs of $C^{\infty}$ funcitons on the real line. A derivation of $\mathcal{O}_{\mathbb{R},0}$ is a $\mathbb{R}$-linear map $\partial:\mathcal{O}_{\...
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1 vote
1 answer
114 views

Maximal Cohen-Macaulay modules of full support, over non-artinian local Cohen-Macaulay rings, are faithful?

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of positive dimension. Let $M$ be a finitely generated maximal Cohen-Macaulay module i.e. $\operatorname{depth} M=\dim R$. If $\operatorname{Supp}(...
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1 vote
1 answer
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For a Noetherian local ring of positive depth, the maximal ideal can be generated by non-zero-divisors?

Let $(R,\mathfrak m)$ be a Noetherian local ring of positive depth. Then, is it always true that $\mathfrak m$ can be generated by non-zero-divisors ? I.e., can we find non-zero-divisors $x_1,\dots,...
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5 votes
2 answers
147 views

Is a local ring with a certain finite quotient necessarily noetherian?

Let $R$ be a commutative unital ring, and let $p$ be a prime number. Question 1: If $R$ is a local ring (with unique maximal ideal $m$) of characteristic $0$, $p \in m$, and $R/p$ is finite, does it ...
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2 votes
0 answers
30 views

Hypothesis - am I doing this right? (Local, principal ideal ring)

Let $R$ a commutative local principal ideal ring with 1 that is not Artinian. So it's Krull dimension is non zero. Let $P\subsetneq M$ a prime ideal and M the maximal ideal of R, since P and M are ...
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6 votes
1 answer
82 views

Let $A$ be an integrally closed noetherian domain, and $(R, \mathfrak{m})$ local with $A \subseteq R \subseteq K(A)$. Is $R$ a localization of $A$?

Let $K(A)$ denote the fraction field of $A$. For context, I'm trying to prove $A = \bigcap_{\text{ht}(\mathfrak{p}) = 1} A_{\mathfrak{p}}$ for an integrally closed domain $A$, from Atiyah-Macdonald's ...
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  • 1,767
1 vote
1 answer
89 views

ring of formal power series is not always local ring?

Let $R$ be a local ring,then,I know $R[[x]]$ is also local ring. But what about when $R$ is not local ring? For example, I heard $\mathbb Z[a1,a2,a3,a4,a5,a6][[x]] $($a1,a2,a3,a4,a5,a6∈K$:field) is ...
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  • 2,701
0 votes
0 answers
60 views

Preimages of non-zero ideals under Unramified morphism

Let $\varphi:A \to B$ be a flat, unramified be ring morphism between Noetherian, local, normal rings $A,B$. Assume $\mathfrak{a} \subset B$ is a non-zero ideal of $B$. Question: Is it true that if $\...
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