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Questions tagged [localization]

For questions regarding the process, consequences, and stability of localizing algebraic structures such as rings, categories, and modules. Not for use with local topological spaces.

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Localization correspondence

This is taken from Neukirch's algebraic number theory Proposition 12.3. Proposition (12.3). If $a\neq 0$ is an ideal of an order $o$, then: $o/a = \oplus_{p}o_p/ao_p = \oplus_{a\subseteq p}o_p/ao_p$ ...
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Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set

I'm mainly trying to prove that If $0\not \in S\subseteq R$ is a multiplicative subset of a commutative ring $R$ with identity. Then $R-S$ contains a prime ideal. Now, by using Zorn's lemma, ...
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Does $\frac{b}{s} \in S^{-1} I$ directly implies that $b \in I$?

Let $R$ be a commutative ring with identity $1_R$, $0 \not \in S\subseteq R$ be a multiplicative set, and $I\subseteq R$ be an ideal of $R$.Consider the ring of quotients $S^{-1}I$. I was trying to ...
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Prove $R_{\mathfrak p}$ has only one maximal ideal $\mathfrak pR_{\mathfrak p}$

$\mathfrak p$ is prime ideal of commtative ring $R$. localization $R_{\mathfrak p}:=(R-{\mathfrak p})^{-1}R$. We know $\mathfrak pR_{\mathfrak p}=(R-{\mathfrak p})^{-1}\mathfrak p$ is prime ideal of $...
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$R_p$ is a field iff for any $x \in p$ there exists $y \not\in p$ such that $xy = 0$

Show that the localization at p, prime ideal, $R_p$ is a field iff for any $x \in p$ there exists $y \not\in p$ such that $xy = 0$ I know there is a similar question where R is an Noetherian ring, ...
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Question regarding localization of polynomial ring

I made an exercise that went as follows: Suppose $R = \frac{\mathbb{R}[x,y]}{(xy)}$. Define the multiplicative set $$ S = \left\{ 1 + (xy), x + (xy), x^2 + (xy), \ldots \right\}.$$ In the exercise, I ...
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1answer
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What does it mean for localization to be same if multiplicative set is different? $T^{-1}R = S^{-1}R$?

If R is a ring and p is a prime ideal, I was told that somehow $Frac(R/p)$ was the localisation $(R \setminus p)^{-1}R/p$. I thought it was more like the localisation $(R/p \setminus\{p\})^{-1}R/p$ ...
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Integrally closed domains, where is the reduced hypothesis used?

The question is to show, given a local domain $A$, with maximal ideal $(\pi)$, $B$ a domain containing $A$, $S=A-(\pi)$. If $S^{-1}B$ is integrally closed, and $B/\pi B$ reduced, then $B$ is ...
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When is the localization map injective?

Suppose that $R$ is a commutative ring and $S\subset R$ is multiplicatively closed subset, i.e. $1\in S$ and if $a,b\in S$ then $ab\in S$. Consider the natural mapping $\phi:R\to S^{-1}R$ defined by $\...
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Question about the ring $A_P$ of regular functions on a variety $V$

Note: I am doing an introductory course in Commutative Algebra and not Algebraic Geometry; This is just a quick application I am trying to understand. Also, rings are commutative with $1$. When ...
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localization of a polynomial ring at a prime ideal

My question is about the hint of exercise 5.1 of Matsumura: Let $k$ be a field,and $R=k[X_{1},\dots,X_{n}]$ and let $\mathfrak{p}\in \operatorname{Spec} R$. Set $k[X_{1},\dots,X_{n}]/\mathfrak{p}=k[...
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Let $R \subseteq S$ be two local PIDs with the same field of fractions, then $R=S$.

Let $R$ and $S$ be two local principal ideal domains with the same field of fractions $K$. I want to show that if $R\subseteq S$ then $R=S$. I will denote as $\mathfrak{m}_R=(m_R)$ and $\mathfrak{m}...
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Valuation ring and localizations [closed]

Theorem. Let $D$ be an integral domain with identity. The following conditions are equivalent. (1) $D_P$ is a valuation ring for each proper prime $P$ in $D$. (2) $D_M$ is a valuation ring for each ...
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Localisation of $\mathbb{Z}/(p^k)$.

I was looking at the wiki that explains localization. It says that the only way to localize $\mathbb{Z}/(p^k)$ is $\{0\}$. The argument is that the elements of $\mathbb{Z}/(p^k)$ are either units or ...
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When does localisation behave badly?

Localisation seems to be a very useful tool in commutative algebra/number theory, and it seems like in every case I've come across, it behaves incredibly well. By behaves well, I mean that it is ...
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Localization of finitely generated algebra

Let $R$ be a reduced finitely generated algebra over $\Bbb Z$. Let $T$ be a finite set of prime ideals of $R$. Let $S = \bigcap_{p \not\in T} R \setminus p$. 1) Is it true that $A := S^{-1}R$ is ...
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Find maximal ideals in semi local ring with singular

I am trying to analyze the normalization $N$ of a local ring $A_{\mathfrak{m}}$ of a variety with Singular. The normalization (integral closure in its total ring of fractions) is semi-local with its ...
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1answer
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Isomorphism between localizations of graded ring $S_{(P)} \cong [S_{(f)}]_{PS_f \cap S_{(f)}}$

I know that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the prime ideals ...
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1answer
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When does it suffice to show statements about rings only for the local ring after localizing at a prime?

I'm learning commutative Algebra with the book from Eisenbud. But I'm having trouble understanding some of his Proofs. Often we have a Statement about a ring R or an R-module M, which we'd like to ...
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localization at one element, regular element

I'm trying to prove that An element $\frac{a}{f^n}$ of $A_f$ is regular iff $\frac{a}{f^n}$ is regular in $A_p$, for all prime ideals $p$ of $A$ such that $f\notin p$. Where $A$ is a unitary ...
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On derived categories of exact categories

The excellent overview paper Exact Categories - Bühler discusses exact categories and all basic definitions surrounding them. In particular section 10 discusses the derived categories of exact ...
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For $m \in \text{max} \ R[X]$ and $f \in R[X] \setminus m, \ R[X]_f$ is not a local ring

Let $R$ be a commutative ring with $1$ and $m$ be a maximal ideal of $R[X]$. Let $f \in R[X]$ such that $f \notin m$. Then I want to show that $R[X]_f$ is not a local ring, where $R[X]_f$ is the ...
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Minimal prime of a ring is contraction of some prime ideal of any extension of the ring. [duplicate]

Let $S$ be a commutative ring with $1$ and let $R$ be any subring of $S.$ Let $p$ be a minimal prime of $R.$ Then how can I show that there is a prime ideal $q$ of $S$ whose contraction is $p,$ i.e., $...
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A question about $S^{-1}R\otimes_R M\cong S^{-1}M$ as $S^{-1}R$-module

I have found this online notes from Columbia University. However, I have a question (maybe very silly) about the proof. It says that by the universal property, the map $f'$ induces a unique $S^{-1}R$...
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What is $S^{-1}N$ in $S^{-1}M$?

I have some problem in understanding the following. Let $M$ be an $A$ module and $S \subset A$ be any multiplicative closed set. Now consider the canonical $A$ linear map $\phi : M \to S^{-1}M,$ i.e., ...
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What is the induced $G$-action on the localization $S^{-1} A$?

This is a somewhat trivial question, but Atiyah-MacDonald doesn't quite specify and I can't find a reference at the moment. Let $A$ be a ring with an action of a group $G$, and $S \subset A$ a ...
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Are the following definitions of a local property equivalent?

I have seen two different definitions of a local property of rings. $P$ is a local property of rings if $P(A)$ is equivalent to $P(A_{\mathfrak p})$ for all prime ideals $\mathfrak p$. $P$ is a local ...
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Global sections of pullback of $G$-equivariant $D_Z$-modules

Let $G$ be a semi-simple complex algebraic group with lie algebra $\mathfrak{g}$. For a fix Borel subgroup $B$ let $X=G/B$ be the flag variety. Let $i_l,i_r:X \to X \times X=Z$ denote the inclusion of ...
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An isomorphism between the residue field and the fraction field

Let $A$ be a commutative ring with $1_A$. Set $X=Spec(A)$. For $x\in X$, let $j_x$ be the corresponding prime ideal. I have already understood and proved the following claims: 1) $A-j_x$ is a ...
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1answer
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The localization of a commutative ring with unity

Let $S$ be a multiplicative set in a commutative ring $A$ with unity. We shall denote the localization of $A$ at $S$ with $A[S^{-1}]$. Let $T$ be another multiplicative set in $A$ such that $S \...
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Two definitions of a local property of rings and modules.

I saw a definition of a local property of rings in the Stacks project: Link. In the second line, do they mean that $P(R)$ is true if $P(R_{f_i})$ for some $f_i$ or that it is true if $P(R_{f_i})$ for ...
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What is the definition of a localization of a category?

There appears to be a discrepancy in the literature regarding the definition of a localisation of a category. Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms. The classical ...
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1answer
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What is the difference between a directed set and a filtered category?

This may seem like a stupid question, but these two concepts seem to be identified so often that it's just a detail I've overlooked. Apparently a filtered category is a generalisation of a directed ...
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1answer
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Localization of Tensor

This question has been asked many times in math overflow, but I can't find justification in any of them in a particular step. I will be using the following theorems. Theorem 1: Let $A,B$ be rings and ...
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TDOA: does increasing amount of receivers when applying TDOA methods increase precision?

NOTE: i wanted to tag the question with TDOA precision multilateration but i dont have the ...
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1answer
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Existence of an epimorphism of the localization to a quotient ring

I'm reading a book about Commutative Algebra from my library and I'm stuck in a problem about localization. Here I put the statement and my attemps. Let $M$ be a maximal ideal of a domain $R$ and let ...
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1answer
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Example of a non-zero module that has no associated primes

I'm currently reading up on Associated Primes and localization. I came across the following theorem. Let $M$ be an $R$ module. If $M = 0$ then $Ass(M)$ is empty. The converse is true if $R$ is a ...
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If the images of $m_1, \dots, m_r$ in $S^{-1}M$ generate the module, then so do their images in $M_f$, for some $f \in S$.

Suppose $R$ is a ring and $M$ is a finitely generated module such that the images of $m_1, \dots, m_r$ in $S^{-1}M$ generate $S^{-1}M$ as a $S^{-1}R$-module. Then prove that the images of $m_1, \dots, ...
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Example of a characteristic zero local ring with a quotient of positive characteristic

This question was featured on a qualifying exam at my university: What's an example of a commutative local ring $R$ of characteristic zero, with a non-maximal prime ideal $P$ such that the ...
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On associated prime ideals of reduced commutative rings $R$ [closed]

How can I prove the following statement? If $R$ is a reduced Noetherian commutative ring, then every associated prime ideal of $R$ is a minimal prime ideal of $R$.
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On commutative reduced local rings [closed]

If $R$ is a reduced local commutative ring with maximal ideal $\mathfrak m$ which is also an associated prime ideal ($\mathfrak m=\operatorname{Ann}_R(a)$ for some $a\in R$). How can I prove that $R$ ...
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Consider $\{D_i:i\in I\}$ as a collection of multiplicative closed subsets of a commutative ring $R$ and the localisation over $M$. [closed]

Consider $\{D_i:i\in I\}$ as a collection of multiplicative closed subsets of a commutative ring $R$. Show that the following statements are equivalent (1) If $D^{-1}_iM=0$ for every $i\in I$ and ...
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Let $Q$ be a prime ideal of an integral domain $A$. Is $A[X]_{Q[X]}$ some localisation of $A_Q[X]$ ?

Let $A$ be an integral domain and $Q$ be a prime ideal of $A$. Then $Q[X]$is a prime ideal of $A[X]$. My question is : Is $A[X]_{Q[X]}$ some localisation of $A_Q[X]$ ?
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1answer
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$A$ is a domain, $Q$ a prime ideal of $A$; connection between integral closedness of $A_Q$ and $A[X]_{Q[X]}$

Let $A$ be an integral domain and $R=A[X]$. Let $Q$ be a prime ideal of $A$ and let $P=Q[X]$. If $R_P$ is integrally closed (in its own fractions field), then is $A_Q$ integrally closed (in its own ...
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Why is this morphism in the saturation of a localizing set of this category?

I am reading the expository paper here. In particular, I am trying to understand the following proof: Let $\mathcal{C}$ be a category admitting all small coproducts. Let $\Sigma$ be a set of morphisms ...
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1answer
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Deciding whether the maximal ideal $\mathfrak{m}R_\mathfrak{m}$ is generated by two elements

Let $\phi$ be a ring homomorphism $\mathbb{C}[x,y,z] \to \mathbb{C}[s,t]$ such that $\phi(x) = s,\ \phi(y) = st,$ and $\phi(z) = t^2.$ Let $R = \operatorname{Im}\phi.$ Let $(a,b,c)\in\mathbb{C}^3$ and ...
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1answer
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On the two definitions of Rings of Quotients in T.Y.Lam

In the book "Lectures on Modules and Rings" by T.Y.Lam there are two definitions: ((8.2) Definition, T.Y.Lam.) We say that $N$ is a dense submodule of $M$ (written as $N\subseteq_d M$) if, for any $...
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1answer
57 views

Examining $\mathbb{Z}[\sqrt{-3}]_{\mathfrak{p}}\subset \mathbb{Z}[\frac{1+\sqrt{-3}}{2}]_{\mathfrak{p}}$

The problem is to show that if $\mathfrak{p}\ne 0$ is a prime ideal of $\mathbb{Z}[\sqrt{-3}]=R$, and we denote $S=\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ then we have $R\subset S$ and for the ...
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2answers
60 views

Show that $z - xy$ is in the ideal $I = \langle 2x-1, 3y-1, 6z-1 \rangle$ in $\mathbb{Z}[x,y,z]$

This is inspired by an exercise in Eisenbud's text on commutative algebra. In the text, he gives the following alternative characterization of localization: for any commutative ring $R$ with unity and ...
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140 views

Inclusion of Reflective Subcategory Creates Limits

$ \newcommand{C}{\mathcal{C} } \newcommand{D}{\mathcal{D} } \newcommand{I}{\mathcal{I} } $ (Full) subcategory $\D$ of the category $\C$ is called reflexive if there exists a localization functor $L ...