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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.

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

Also for relationships to Spec and quasi-coherent sheaves.

1176 questions
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Why is the localization at a prime ideal a local ring?

I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is the localization of the ring $ A $ with respect to…
Bryan
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41
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3 answers

Why is the localization of a commutative Noetherian ring still Noetherian?

This is an unproven proposition I've come across in multiple places. Suppose $A$ is a commutative Noetherian ring, and $S$ a multiplicative subset of $A$. Then $S^{-1}A$ is Noetherian. Why is this? I thought about taking some chain of…
Buble
  • 1,659
39
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An integral domain $A$ is exactly the intersection of the localisations of $A$ at each maximal ideal

This result appears to be ubiquitous as an algebra exercise. How do you prove this result? Let $A$ be an integral domain with field of fractions $K$, and let $A_{\mathfrak{m}}$ denote the localisation of $A$ at a maximal ideal $\mathfrak{m}$…
user118000
  • 391
26
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5 answers

Localisation is isomorphic to a quotient of polynomial ring

I am having trouble with the following problem. Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$. I just want a hint. Basically, I've been looking…
johnq
  • 261
22
votes
2 answers

Motivation behind the definition of localization

What is the motivation behind definition of localization of rings? From where does the term "localization" come from? Why is the equivalence relation between the ordered pairs $(m,u),(m',u')$ with $ m,m' \in M$ and $u,u' \in U$ is defined as…
Mohan
  • 15,494
21
votes
3 answers

Is the quotient ring of a PID a PID?

Let $A$ be a commutative ring and $S$ a multiplicative closed subset of $A$. If $A$ is a PID, show that $S^{-1}A$ is a PID. I've taken an ideal $I$ of $S^{-1}A$ and I've tried to see that is generated by one element; the ideal $I$ has the form…
Sadi
21
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2 answers

Localization commutes with Hom for finitely-generated modules

Let $M,N$ be $R$-modules with $M$ finitely generated and let $S\subseteq R$ be multiplicatively closed. Then there exists a module isomorphism $$S^{-1}\text{Hom}_R(M,N) \xrightarrow{\sim} \text{Hom}_{S^{-1}R}(S^{-1}M,S^{-1}N).$$ As homework, I…
MadPidgeon
  • 323
20
votes
1 answer

Showing that localization is an exact functor

I'm again in this awfully familiar situation where I'm struggling to prove simple statements mostly because I have no idea how a template of a proof should look like in this specified context. I'm trying to prove these two statements: Given a…
Saal Hardali
  • 5,059
17
votes
5 answers

Localization at a prime ideal in $\mathbb{Z}/6\mathbb{Z}$

How can we compute the localization of the ring $\mathbb{Z}/6\mathbb{Z}$ at the prime ideal $2\mathbb{Z}/\mathbb{6Z}$? (or how do we see that this localization is an integral domain)?
user9459
17
votes
2 answers

Exactness of sequences of modules is a local property

It's well known, that passing to modules of fractions is exact, i.e. if $$M'\xrightarrow{f} M\xrightarrow{g} M''$$ is an exact sequence of $A$-modules ($A$ being a commutative ring with unity), then for every multiplicative subset $S\subset A$, the…
Ben
  • 6,587
14
votes
4 answers

Localization commutes with arbitrary direct sums

Let $M_i$ be a arbitrary colection of $A$-modules and $S$ a multiplicative subset of $A$. I want to show that $$S^{-1}\left(\bigoplus_i M_i\right)\cong \bigoplus_i S^{-1}M_i$$ as $A$-modules and as $S^{-1}A$-modules. I know how to explicitly write…
Gabriel
  • 4,761
14
votes
1 answer

Showing that the fiber of a morphism of schemes over a point is homeomorphic to the preimage

Suppose I am given a morphism of schemes $f: X \longrightarrow Y$ and suppose that for any point $y \in Y$, $\kappa(y)$ denotes the residue field. I would like to show that the fibered product $$X \times_{Y} \text{Spec}(\kappa(y))$$ is homeomorphic…
Luke
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14
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2 answers

Localization of a local ring

If $A$ is a local ring, $\mathfrak{m}$ is its maximal ideal, then is $A_{\mathfrak{m}}\cong A$ ? That's, in my opinion, because every denominator is invertible. Am I right?
aleio1
  • 1,065
13
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1 answer

There isn't a "going in-between" theorem for integral ring extensions, is there?

Let $A\subset B$ be an integral extension of commutative, unital rings. We have the well-known "incomparability", "lying-over", and "going-up" theorems (5.9-5.11 in Atiyah-Macdonald, or see here). Lying-over asserts that every prime of $A$ has a…
Ben Blum-Smith
  • 21,560
13
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3 answers

Localization commutes with quotient

Suppose $R$ is a commutative ring and $D$ a multiplicatively closed subset. I'd like to show via universal properties that if $\mathfrak{a} \triangleleft R$ is an ideal, then $D^{-1}R/D^{-1}\mathfrak{a} \cong \bar{D}^{-1}(R/\mathfrak{a})$, where…
Eric Auld
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