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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$ \mathfrak mR_{\mathfrak m} $-primary ideal is the localization of some $\mathfrak m$-primary ideal?

Let $\mathfrak m$ be a maximal ideal of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R_{\mathfrak m}$ such that $\mathfrak m^n R_{\mathfrak m}\subseteq J \subseteq \mathfrak mR_{\...
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Infinite product of fields with non noetherian localization

Let $F$ be a field, and let $R=\prod_{i=1}^\infty F$. In wikipedia, and more specific: There is a claim which I suspect is wrong and here is why: Let $m\triangleleft R$ be a maximal ideal. Hence, ...
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Is the category of modules over the localization ring an example of a localization of a category?

Consider r to be a non-unit element of a ring R and S the subset of R consisting of its natural powers. Wikipedia states that the categorical localization of the category of modules over R with ...
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Isomorphism in the quotient ring of a localization

Let $R$ be a ring with unity and suppose $I$ is a maximal ideal of $R$ such that $M = R \setminus I$ is a right denominator set for $R$. Is it true that $R_M/I_M \simeq R/I$, where $R_M$ is the right ...
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4 votes
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If localization ring is a domain, the ring doesn't have to be a domain [duplicate]

Let $A$ be a ring and let $A_{\frak{p}}=S^{-1}A$ with $S=A-\frak{p}$. I know that if $A_{\frak{p}}$ is a domain for every $\frak{p}$ prime ideal that doesn't mean that $A$ is a domain. However, I have ...
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Does localisation (of modules) cancel.

If I have two modules $R$ and $M$ and I take the localisation of both then do we have, $$\frac{S^{-1}R}{S^{-1}M}\cong\frac{R}{M}$$
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Why is this residue field equal to the field of fractions in this case? [duplicate]

I have the following question. If $p$ is a prime ideal then by localization with the multiplicative set $S:=R\setminus p$ we get the local ring $R_p=:S^{-1}R$ and a maximal ideal $pR_p$. By ...
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3 votes
1 answer
170 views

Exercise about localization and monic polynomial

Please help me solve this exercise, Suppose $R$ is an integral domain and $S$ is a multiplicatively closed subset. Prove that if every element of $R_{S}$ is a root of a monic polynomial of $R[x]$ ...
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Localization an ideal by maximal ideal [duplicate]

I try to show this, suppose $R$ be a commutative ring with $1$ and $A$ be an ideal in $R$. Show if $A_{M}=0$ for all maximal ideal $M$, then $A=0$. My idea this, If $a\in A$ then for every maximal ...
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How to compute the quotient and localization of the monoid algebra $kG$ for a field $k$

I am given that $k$ is a field and $G$ is the monoid consisting of all monomials $X^iY^j$, where $j$ is between $0$ and $3i$. I am trying to compute the quotient of the monoid algebra $kG$ by the ...
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Describe the localization

I recently got a rather vague question which stated Describe the localization $(\mathbb{Z} / 15\mathbb{Z})_{(5)}$ (away from the prime ideal $(5)$). I know now that this localization is somehow ...
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1 answer
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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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$A_p \otimes A/q = \frac{A_p}{qA_p}$

Let $A$ be a commutative ring and $p$, $q$ prime ideals such that $q \subseteq p$. Show that $A_p \otimes A/q = \dfrac{A_p}{qA_p}$. Here's what I have done - Let $S = A \setminus p$. Then $A_p = S^{-1}...
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$K[t^2,t^3]_{(t^2,t^3)}$ is not regular

I want to show that $K[t^2,t^3]_{(t^2,t^3)}$ is not regular, and to do so I want to show that $dim_F(m/m^2)=2$ for $m=(t^2,t^3).K[z^2,t^3]_{(t^2,t^3)}=(\frac{t^2}{1},\frac{t^3}{1})\subseteq K[t^2,t^3]...
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$A_f \otimes_A A_g = A_{fg} $

A is a commutative ring. $f,g\in A$. Then I want to show $A_f \otimes_A A_g = A_{fg}$. I tried proving it by constructing a bilinear form from $A_f \times A_g$ to $A_{fg}$. But I could not figure ...
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4 votes
1 answer
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Proof of local definition of Dedekind Domain without using unique factorization of ideals

We say that an integral domain $A$ is a Dedekind domain if: $A$ is Noetherian, $A$ is integrally closed, $\dim A = 1$ (in other words, every prime ideal is maximal). I would like to show that ...
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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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1 vote
3 answers
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Localization of $\mathfrak p$ is $\mathfrak p R_{\mathfrak p}$

In the setting of this question it is mentioned that Since localization preserves exact sequences we have that the exact sequence $$0 \to \mathfrak p \to R \to R/\mathfrak p \to 0$$ becomes $$0 \to \...
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1 answer
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Nonzerodivisor of a module under localization

If $x_1,x_2,...,x_r\in R $ is an $M$-sequence (i.e. a regular sequence on $R$-module $M$), then $x_1^t,...,x_r^t$ is an $M$-sequence. This is corollary 17.8 from Eisenbud's commutative algebra. The ...
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Finitely generated module over Noetherian ring, all whose localizations at associated primes of the ring is $0$, is a torsion module?

Let $M$ be a finitely generated module over a commutative Noetherian ring $R$ such that that $M_P=0$ for every associated prime $P$ of $R$. Then, is it true that for every $m\in M$, there exists a non-...
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How to see this assertion about coordinate rings and a localisations

I am studying 'The Homogeneous Coordinate Ring of a Toric Variety' by David Cox, and in the proof of his theorem 2.1 he defines $U_\sigma := \{x\in \mathbb{C}^{\Sigma(1)} : x^{\hat{\sigma}}\neq0\}$ ...
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2 votes
1 answer
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Equivalent statements of a finitely generated module being locally free

Let $M$ be a finitely generated $R$-module. Prove that the following conditions on $M$ are equivalent: (a) $M$ is locally free over $R$ (i.e. $M_m$ is free over $R_{m}$ for all maximal ideals $m\...
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0 answers
48 views

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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1 vote
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72 views

$M_p \cong N_p$ for all prime ideals $p$, but $M \not \cong N$

I am asked to find an example of a ring $R$ and $R$-module $M$ and $N$ such that $M_p \cong N_p$ for all prime ideal $P$ in $R$ but $M$ is not isomorphic to $N$. My idea is as follows: Recall that if $...
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2 answers
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If $ S \subset A^{\times}$, then $S^{-1}A \simeq\ ?$ (A question related to localization of rings)

This question was asked in my quiz of commutative algebra and I was unable to solve this particular question. So, I am posting it here. If $S\subset A^{\times} $ (set of all units), then $S^{-1} A \...
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1 vote
0 answers
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Question about Anderson Localization and a specific theorem (RAGE)

I have a question with a theorem which appears in a text called “Invitation to Random Schrödinger Operators”, in unit 7. Theorem 7.7. Let $H$ be a selfadjoint operator on Hilbert Space, take $\psi \...
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Is the localized module of derivations the module of derivations of the localized algebra?

Let $X$ be an affine algebraic variety and $A=\mathbb{C}[X]$ its coordinate ring. The vector space $$ \mathrm{Der}_{\mathbb{C}}(A) = \{ D: A \rightarrow A \mid D \text{ is linear and } D(ab)=D(a)b+aD(...
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Geometric meaning of localization with an example

My task is to understand the following example: Let $R=\mathbb{R}[x,y]/(xy).$ Let $p=(x-1)$ be the ideal of $R$. Show that $(x-1)$ is a maximal ideal of $R$ and deduce that the localization at this ...
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Geometric intuition about and examples of locally cyclic modules

A collection $S_1,\dots,S_n$ of monoids (which we always assume contain $1$) in a commutative ring $A$ are said to be comaximal when any $\langle s_1,\dots,s_n\rangle=1$ for any choice of $s_i\in S_i$ ...
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Question about the restriction $\tau_n>0$ in the definition of local martingales or localization of processes

I have a question while reading the proof of Lemma 3 from the following link: https://almostsuremath.com/2009/12/24/local-martingales/ The Lemma states that a nonnegative local submartingale is a ...
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1 vote
1 answer
43 views

How many elements does $\mathbb{Z}[1/6]/(2020)\mathbb{Z}[1/6]$ have?

Consider the $\mathbb{Z}$-module $\mathbb{Z}[1/6]$ and the module $(2020)\subset \mathbb{Z}$. In an algebra exercise from last year, it was asked how many elements the quotient $\mathbb{Z}[1/6]/(2020)\...
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1 vote
1 answer
59 views

Two affine varieties which are isomorphic, but their projectivisations are not

I am learning Algebraic Geometry and came across the following question: Show $V(y-x^3) \cong V(y-x^2)$ as affine varieties in $\mathbb{A}^2$. Prove that their projectivisations in $\mathbb{P}^2$ are ...
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3 votes
1 answer
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Let $M$ be an $R$-module. If $I \subseteq \mathfrak{m}$ implies $M_\mathfrak{m} = 0$ for all max ideals $\mathfrak{m} \subseteq R$, show $IM = M$

Suppose we have a ring $R$ and an $R$-module $M$. Suppose we have an ideal $I\subseteq R$ such that for all maximal ideals $\mathfrak{m} \supseteq I$ we have $M_\mathfrak{m} = 0$. I need to show that ...
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1 vote
1 answer
74 views

Very basic question about localizations

Let $A$ be a ring, $S$ a multiplicatively closed subset. Is it true that $\frac a1\in S^{-1}A$ is a non zero-divisor if and only if $a\in A$ is? I would say yes: for, if $\frac a1$ is a zero-divisor ...
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0 votes
2 answers
92 views

Question about universal property of the localization

I have this doubt, that arise quite often when I work with localizations. Let $S\subset A$ be a multiplicatively closed set of a ring, and let $i:A\to S^{-1}A$ be the canonical homomorphism. Suppose ...
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1 vote
1 answer
50 views

localization in the D + M construction

Let $T$ be an integral domain, and let $K$ be a subfield of $T$ and $M$ a maximal ideal of $T$ such that $T = K + M$. Let $D \subseteq K$ be an integral domain, and set $R = D + M$. My question is: if ...
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-1 votes
1 answer
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$R_P\cong \mathbb{R}[x]_{(x-1)}$

Let $R=\mathbb{R}[x,y]/(xy)$ and $P\subset R$ be the ideal generated by $x-1$. Show that $$R_P\cong \mathbb{R}[x]_{(x-1)},$$ where $R_P$ is the localization of $R$ at $R\setminus P$ (since $R/P$ is a ...
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2 votes
0 answers
39 views

Another proof that minimal primes are made of zero-divisors

Let $A$ be a ring, let $D$ be the set of the zero-divisors and $S:=A-D$. Show that any minimal prime ideal of $A$ is contained in $D$. I know that this question has already been asked, but (in the one ...
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1 vote
1 answer
27 views

Characterization of maximal multiplicatively closed subsets

Let $A$ be a non-zero ring, and let $\Sigma $ be the set of the multiplicatively closed subsets properly contained in $A$. Show that $\Sigma$ admits a maximal element respect to inclusion. Then prove ...
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0 votes
0 answers
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Isomorphism between localizations (2) [duplicate]

Let $f:A\to B$ be a ring homomorphism, and let $S\subset A$ be a multiplicatively closed set. Denote with $T$ the image of $S$ in $B$. Show that $S^{-1}B\cong T^{-1}B$ as $S^{-1}A$-modules. First I ...
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0 votes
0 answers
44 views

Isomorphism between localizations

Let $S,T$ be two multiplicatively closed subsets of a ring $A$. Let $U$ be the image of $T$ in $S^{-1}A$. Show that $(TS)^{-1}A\cong U^{-1}(S^{-1}A)$. $S^{-1}A\to (TS)^{-1}A:\frac as\mapsto \frac as$ ...
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1 vote
0 answers
68 views

Example of a certain ring

I have to find a ring $A$ that is not an integral domain, but such that $A_p$ is an integral domain for every prime ideal $p\subset A$. Set $A:=\mathbb Z/6\mathbb Z$. Its only (prime) ideals are $p:=3\...
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0 votes
0 answers
21 views

Exercise about saturations of a ring

Let $A$ be a ring, $S$ any multiplicatively closed subset. For an ideal $I$, let $S(I)$ denote the contraction of $S^{-1}I$ in $A$. If $I$ has a primary decomposition, show that the ideals of the form ...
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2 votes
0 answers
40 views

Localization of cocomplete categories and right orthogonality: does the equivalence always hold?

In Handbook of Categorical Algebra, Volume 1: Basic category theory, Borceux proves the following (around theorem 5.4.7 page 198). Definition. An object $x$ of a category $\newcommand{\cC}{\mathsf{C}}\...
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3 votes
2 answers
127 views

Exercise with localization and minimal primes

Let $A$ be a ring and $p\subset A$ be a prime ideal. Call $f$ the canonical map $A\to A_p$, and set $I:=\operatorname {ker }f$. Show that $I\subseteq p$ and that $\sqrt I=p$ if and only if $p$ is ...
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  • 2,443
0 votes
0 answers
26 views

Question about ideals of the form $(0:a)$

Let $A$ be a ring. For any $a\in A$, define $(0:a):=\{x\in A:ax=0\}$. Now let $S\subseteq A$ be a multiplicatively closed set; if $S\cap (0:a)=\emptyset$, does $S^{-1}(0:a)=(0:\frac a1)$? ($\subseteq$)...
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4 votes
1 answer
99 views

Problem 3.24, Commutative Algebra Atiyah & Macdonald

I'm getting stuck in understanding the solution of the following exercise in Commutative Algebra text by Atiyah & Macdonald: Let $(U_i)_{i \in I}$ be a covering of $X = Spec(A)$ by basic open sets....
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0 answers
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Question on 1-dimensional Noetherian local domains

Suppose that $A$ is a Noetherian domain of dimension $1$, in which every primary ideal is a power of some prime ideal. Being $A$ a $1$-dimensional domain, the only non-maximal prime ideal is $(0)$. ...
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4 votes
1 answer
73 views

Local Global principle and Modules

Let $V$ be an A-module over a commutative ring $A$. (a) Let $x,y \in V$. Then $x=y$ $\Leftrightarrow \frac {x}{1}=\frac{y}{1}$ in $V_M$ for all $M\in Spm A$. If I have $x=y$ then for each $m\in M$ I ...
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1 vote
1 answer
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Localization and Field [duplicate]

Let K be a field, I an infinite indexed set and A be the product ring $K^I$. For every $p\in Spec A$, prove that the localization $A_p$ is a field. In particular, $p \in Spm A$. This question was ...
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