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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A problem related to localization

Let $A$ be integrally closed (not necessary be Dedekind), $K=\operatorname{Frac}(A),L/K$ Galois, $B$ is the integral closure of $A$ in $L,p$ is a maximal ideal in $A$. We know that the galois group $G$...
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Localization with respect to set that is NOT mutliplicatively closed

All rings are assumed to be commutative with unity. Usually denoted $R$ Intro: I understand that there are two definitions of localization (or possibly more, but these two are somewhat canonical for ...
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Laurent polynomials $ \mathbb C[t,t^{-1}]$ is the localization of $\mathbb {C}[t].$

I want to prove this question: Show that the ring of Laurent polynomials $ \mathbb C[t,t^{-1}]$ is the localization of the polynomial ring $\mathbb {C}[t].$ Localization is defined as follows: Let $R$ ...
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how to account for change in height in a 2d positioning system

I am working on DW1000 based positioning system and have so far achieved great positioning using 2 anchors (ignoring -ve x to avoid 2 points with same distance) .But this system is unable to handle ...
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Description of submodules under localization

I'm have being reading Einsenbud's conmutative algebra. And wanted to do exercise 2.9. enter image description here And while I believe that I understand the idea of the problem, I'm not sure how to ...
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Localization at a prime ideal

Let $k$ be a field with characteristic 0. Let $R=k[x]$ consider the module and $U=\{1,x,x^2,\dots\}$. Compute $R[U^{-1}]$ and $R[(R\setminus U)^{-1}]$. After some calculations an interpretation I ...
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Set of units in localization of $R$

Let $R$ be a ring, and let $P$ be a prime ideal of $R$. Let $S = R \setminus P$, and set $R_p = S^{-1} R$. What are the units in $R_p$? I'm trying to prove that $\frac{a}{b}$ is an unit if and only if ...
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$M\subset N$ for $R$-modules $M,N$ if $S_{\mathfrak m}^{-1}M\subset S_{\mathfrak m}^{-1}N$ for all maximal ideals $\mathfrak m\subset R$?

Consider the following proposition (with proof) taken from S. Lang's "Algebraic Number Theory": Proposition $\mathbf{18}$. Let $A$ be a Dedekind domain and $M,N$ two modules over $A$. If $\...
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Finding $M[U^{-1}]$

I've been studying localizations, and was given the next excercise: Let $k$ be a field with characteristic 0. Let $R=k[x]$ consider the module $M=R/(x^4-x^2)$ and $U=\{1,x,x^2,\dots\}$. Compute $M[U^{-...
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Is the spectrum of a stalk a subscheme?

Given a scheme $X$ and a point $p \in X$, I know that $\text{Spec}(\mathcal{O}_{X,p})$ consists of the point $p$ together with all generic points of irreducible closed subsets containing $p$. Can we ...
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Relationship Between Power Series ODE Solution Techniques?

When solving an ODE via a power series at an ordinary (nonsingular) point, the initial guess is $y = \sum_{n = 0}^\infty a_n x^n$. When solving an Euler ODE, the second order equation $x^2 y'' + pxy' +...
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Integral closure in an extension ring

I'm trying to prove the following statement: Let $R$ be an integral domain and let $S\subset R$ be a multiplicative closed subset. If $R\subset R'$ is an extension ring of $R$ with $\overline{R}$ the ...
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$K$-derivation $\delta_\alpha: N^{-1}R[x]\to N^{-1}R[x]$ with $\delta_{\alpha}(x)=\alpha$

I'm reading Kähler Differentials written by Ernst Kunz. There is an exercise in its first chapter: Let $R$ be a $K$-algebra and $\delta:R\to R$ a $K$-derivation. Let $N\subset R[x]$ be ...
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$\mathbb{Q}$-derivation of a minimal prime ideal

I'm reading "Kähler differentials" written by Ernst Kunz. In the first chapter there is an exercise: If $R$($\supset \mathbb{Q}$) is a ring, for each $\mathfrak{p}\in Min(R)$ and each ...
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When localizing an ideal in a ring of dimension 1, how do you show that you get this?

Let $A$ be a ring of dimension $1$ and let $J$ be an ideal of $A$ that can be factored as a product of maximal ideals $J = P_1^{a_1} \dotsm P_s^{a_s}$. Let $M$ be any maximal ideal of $A$. Then $J_M = ...
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Why are the prime ideals of $R_f$ exactly the prime ideals of $R$ not containing $f$?

Let $R$ be a ring and $D_f=\{\text{primes ideals in } R \text{ not containing } f\}$ be a basic open set. Let $R_f$ be the localization of $R$ at $f$, I am trying to show that $D_f=\operatorname{Spec}...
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Question on notation in Eisenbud's Commutative Algebra With a View Toward Algebraic Geometry Exercise 2.19(b)

I will restate the question here: Let $R$ be a ring and let $M$ be an $R$-module. Suppose that $\{f_i\}$ is a set of elements of $R$ that generate the unit ideal. Prove: If $m_i\in M[f_i^{-1}]$ are ...
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$A$ ring, $\mathfrak{m}$ maximal ideal - then $\mathfrak{m}/\mathfrak{m}^2$ is isomorphic to $\mathfrak{m}A_\mathfrak{m}/\mathfrak{m}^2A_\mathfrak{m}$

According to lemma 4.2.3 of Qing Liu's book on Algebraic Geometry, the canonical homomorphism $\mathfrak{m}/\mathfrak{m}^2 \to \mathfrak{m}A_\mathfrak{m}/ \mathfrak{m}^2 A_\mathfrak{m}$ is an ...
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Polynomial expressions in localized ideal

Let $A$ be a commutative ring and $S\subset A$ a multiplicatively closed subset. Fix a polynomial $f\in A[X]$ and define $I$ to be the ideal generated by elements of the form $f(a)$ with $a\in A$. My ...
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Localization of a principal ring is principal?

I'm trying to prove that the localization $S^{-1}A$ is a principal ring. Here, $A$ is a principal ring and $S$ a multiplicative subset of $A$. Furthermore we assume $0 \notin S$. The term principal ...
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$R$-module with local endomorphism ring is also an $R_P$-module for some $P\in\operatorname{Spec} R$

Let $R$ be a commutative ring, and suppose that $U$ is an $R$-module with local endomorphism ring. (In particular, note that $U$ is indecomposable.) Consider the ring morphism $f:R\to\operatorname{End}...
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$m ∈ M$ such that the image in $M_{\mathfrak{p}}/\mathfrak{p}M_{\mathfrak{p}}$ is not zero

Let $M$ be a finitely generated $A$-module. For every prime ideal $\mathfrak{p}\in \operatorname{Spec}{A}$, define $$μ(\mathfrak{p},M)=\dim_{k(\mathfrak{p})}(M_{\mathfrak{p}}/\mathfrak{p}M_{\mathfrak{...
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Proof verification: Atiyah-Macdonald Exercise 3.7(i)

A multiplicatively closed subset $S$ of a ring $A$ is said to be saturated if: $$ xy\in S \Leftrightarrow x\in S \text{ and } y\in S $$ Exercise 3.7(i): $S$ is saturated if and only if $A-S$ is a ...
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localization tensor product vs tensor product localizations

First of all, I have to say I am aware of this question, which is very close to mine. Hopefully, this will not be a duplicate. Suppose $A$ and $B$ are algebras over some ring $C$ and let $r$ be a ...
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Local property of localization and tensor product

Let $M$ be an $R$-module, and let $U\subset R$ be a multiplicatively closed subset. Let $M_U$ be the localization of $M$ at $U$, and let $R_U$ be the localization of $R$ at $U$. Then we have the "...
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A typo in Proposition 2.2, Eisenbud commutative algebra?

I am trying to understand the formulation of Proposition 2.2 in Eisenbud's "Commutative algebra with the view...". Here is the place that is not clear to me Proposition 2.2. Let $\varphi: R\...
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When is an integral domain a localization of a proper subring?

Given an integral domain $R$, is $R$ a localization of a proper subring if and only if there is some unit $u \in R^{\times}$ for which $u^{-1} \notin \mathbb{Z}[u]$, where $\mathbb{Z}[u]$ denotes the (...
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Computing localizations in $\mathbb{Z}/n\mathbb{Z}$

I am trying to find a way to easily compute localisations of the ring $\mathbb{Z}/n\mathbb{Z}$ ($n>1$). Is there any general result for this? I found here that when the multiplicative subset is $S=\...
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$O_{K,\mathfrak{p}} = \Bbb{Z}[a]_{(p,\pi)}$ when $O_K$ is non-monogenic

This is a fun exercice I came with. Let $K$ be a number field, $O_K$ non-monogenic and $\mathfrak{p}$ a prime ideal above a bad prime $p$ then how would you show that for some $a\in O_K,\pi\in \Bbb{Z}[...
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Show that the following localization ring has one maximal ideal

I was given the following question: $R=\mathbb{Z}$ and $S$ contains all elements not divisible by $p$ (which is a given prime) Prove that $S^{-1}R$ has only one maximal ideal I tried assuming it has ...
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the number of distinct subrings of $\mathbb{Q}$ is infinite without using infiniteness of prime numbers

I know all subrings of $\mathbb{Q}$ is {$\mathbb{Zs}$|$\mathbb{S}$ is complement of union of $\mathbb{pZ}$ where $p$ is a prime}. If the number of prime numbers is finite, the number of distinct ...
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Can the category of S-local objects be reflective but not a localization by S?

This question has been answered in a MathOverflow cross-post. Suppose $\mathcal C$ is a category and $S \subseteq \operatorname{Mor}(\mathcal C)$ is some collection of morphisms in $\mathcal C$. Let $\...
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Atiyah-Macdonald: Exercise 3.5

Let $A$ be a ring. Suppose that, for each prime ideal $\mathfrak{p}$, the local ring $A_{\mathfrak{p}}$ has no nilpotent elements $\neq 0$. Show that $A$ has no nilpotent element $\neq 0$. If each $A_{...
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Limits in the localization of a category of fibrant objects

Suppose we have category $\mathcal{C}$ which has the structure of a category of fibrant objects, and suppose we have a functor $F:I\to \mathcal{C}$ with a limit $\lim F$ in $\mathcal{C}$. If we have ...
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Finite free resolutions of modules over coordinate ring of smooth elliptic curve over $\mathbb{C}$

I believed for a while that the reason you can't find a finite free resolution (with the free modules finitely generated) of $(x - 1, y - 1)$ as a module over $\mathbb{C}[x, y]/(y^2 - x^3)$ is that $y^...
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The ring homomorphism between a commutative ring $R$ and its localization $S^{-1}R$ is invertible on $S$

Let $R$ be a commutative ring, and $S$ be a multiplicative subset, let $S^{-1}R$ be the localization, and defines a ring homomorphism $l: R\to S^{-1}R$ such that $l(a)=\frac{a}{1}$. Can someone tell ...
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Prime ideals of localization of absolutely flat modules

There is an exercise (chapter 3, Exercise 10, page number 44) in atiyah mc Donald. $A$ is absolutely flat iff $A_m$ is a field for each maximal ideal $m$. I actually proved this statement and it is ...
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Give an example of a local domain with principal maximal ideal, which is not a valuation ring

Give an example of a local domain with principal maximal ideal, which is not a valuation ring. If we had Noetherianness on top of it, we'd have that it's a DVR; but without it, I have no idea how to ...
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Adjoining an inverse to a central element of an algebra

Given a (not necessarily commutative) algebra $A$, and a central element $c \in A$, is it always possible to enlarge $A$ to a an algebra $A'$ in which $c$ is invertible? I guess one can take a set of ...
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If $R$ is not integral domain, and $P$ is prime in $R$, is $P^e$ not equal to $PR_P$?

In Dummit & Foote, Proposition $46$(2) on page $718$ says If $R$ is an integral domain, then $R_P$ is an integral domain. The ring $R$ injects into the local ring $R_P$, and, identifying $R$ with ...
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Localization is finitely generated implies, that multiplicative group only contains invertible elements

I am currently stuck on this problem: "Let $R$ be an integral domain, let $0 \notin S \subseteq R$ be a multiplicative group of $R$. If $S^{-1}R$ is finitely generated as a $R$-module, then every ...
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Show that the localization $\mathbb{R}[x]_{(x-1)}$ is not an integral extension of $\mathbb{R}[x^2 -1]_{(x-1)\cap \mathbb{R}[x^2-1]}$

Let $R' = \mathbb{R}[x]$, $R= \mathbb{R}[x^2 -1] \subset R'$, $P' = (x-1) \subset R'$, and $P = P' \cap R$. The given hint suggested to consider the element $\frac{1}{x+1} \in R'_{P'}$. So the ...
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Localisation of an integral domain

My question started as follows: An intermediate ring $\Bbb Z\subseteq R\subseteq\Bbb Q$ arises as a localisation. In particular it holds that $R=S_{P}^{-1}\Bbb Z$, where $S_{P}$ is the multiplicative ...
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Possible Characterization of localization of $\mathbb{Q}[x, y]$

So I wish to know if it is possible to Characterize the localization of $\mathbb{Q}[x, y]$. The motivation behind this question is that I was asked to determine (True or false) that every subring of $\...
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Localization by a prime ideal

Let $A$ be a commutative ring with identy and let $P\subseteq A$ a prime ideal. We consider $$A_P:=\bigg\{\frac{a}{s}\:\bigg|\; a\in A, s\in A\setminus P\bigg\}.$$ I must prove that $$\frac{a}{s}\;\...
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The contraction of $B_p\cdot q$ is $p\cdot A_p$

I am trying to prove the following: Let $A\subset B$ with $B$ integral over $A$. Let $q\subset q' \subset B$ be prime ideals. If $q^c=q'^c\subset A$ then $q=q'$. Here is my attempt with the points I ...
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Localisation of a polynomial ring

Is the ring $\mathbb{Q}[X,Y,X^{-1}]$ isomorphic to $\mathbb{Q}[X]$ ? I think of the first ring as the localisation of $\mathbb{Q}[X,Y]$ at $S=\{1,X,X^{2}...\}$. In my opinion, the localisation makes $...
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Proving that $R_{(0)}$ is the field of fractions.

Here is the question I want the solution of part $(c)$ in it: Let $R$ be an integral domain with field of fractions $K,$ and let $S \subset R \setminus \{0\}$ be a multiplicatively closed subset. $(a)$...
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Show that if $\mathfrak{p} \subset R$ is a prime ideal, then $S = R \setminus \mathfrak{p}$ is multiplicatively closed. [duplicate]

Here is the question that I want to answer: Let $R$ be an integral domain with field of fractions $K,$ and let $S \subset R \setminus \{0\}$ be a multiplicatively closed subset. Show that if $\...
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localization in 2-categories with 2-morphisms

I know almost nothing about higher categories. In a 2-category, is there a way to localize on a set of 2-morphisms rather than 1-morphisms ?

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