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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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Equality of two completions

I have the following question. Suppose $R$ is Noetherian ring, $I$ is ideal in $R$ and $S$ is multiplicatively closed set. Let $(I^n\colon\langle S\rangle) = \varphi^{-1}(I^nS^{-1}R),$ where $\varphi\...
abcd1234's user avatar
2 votes
1 answer
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An infinite cyclic group has infinitely many irreducible real representations.

I'm trying to show that an infinite cyclic group $G=\langle g\rangle$ has infinitely many non-equivalent irreducible representations over $\mathbb{R}$. I have in mind the following argument, for which ...
F. Salviati's user avatar
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1 answer
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Show that $\mathfrak{m}A_{\mathfrak{m}}=(\mathfrak{m}A_{\mathfrak{m}})^n$

Let $A$ be a unital commutative ring with maximal ideal $\mathfrak{m}$ and let $A_{\mathfrak{m}}$ denote the localization of $A$ at $\mathfrak{m}$. I want to show that $\mathfrak{m}^{n}A_{\mathfrak{m}}...
ephe's user avatar
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5 votes
2 answers
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Is the localization of a zero-dimensional ring a quotient?

If $R$ is any commutative zero-dimensional ring and $m$ is a maximal ideal, then is $R_m$ always naturally a quotient of $R$? In other words, is the natural map $R\to R_m$ always surjective? I was ...
Anon's user avatar
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1 answer
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Does the uniformizer $\pi$ generate the extension $K/T$ where $T$ is the inertia field?

I'm trying to prove the equivalence of the statements (1) $\sigma(x) \equiv x$ mod $\mathfrak{p}^{i+1}$ (that is, $\sigma \in V_i$, the $i^{th}$ ramification group) for all $x\in \mathcal{O}_{K,\...
ljfirth's user avatar
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1 answer
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Krull dimension of modules of tensor products with $\mathbb{Q}$ [closed]

Let $\mathbb{Z}[X_1,...,X_n]$ be the ring of polynomials in $n$ variables over the integers. Let $M$ be a nontrivial finitely generated module over $\mathbb{Z}[X_1,...,X_n]$ which is torsion free as ...
Josh F's user avatar
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Morphic Composition in Localization Categories

In the category localization discussed in the Stacks Project, there is a description in the morphism composition of localized categories as follows: The composition of the equivalence classes of the ...
jhzg's user avatar
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Proving equivalent conditions for an automorphism to lie in the $i^{th}$ ramification group.

I am following these notes on the Kronecker-Weber Theorem and it quotes without proof the following equivalences. Below, $K$ is a number field, $\mathfrak{p}$ is a prime ideal in the ring of integers $...
ljfirth's user avatar
  • 520
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1 answer
33 views

Homomorphisms under localization

I have the following Lemma in my course: Assume $M$ is a finitely presented $A$-module, $S \subset A$ is a multiplicative set. Then there exists a natural isomorphism of $S^{-1}A$ modules: $$ \mathrm{...
Pastudent's user avatar
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3 votes
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Finite type ring homomorphisms and finitely many localizations

Let $\phi : B \to A$ be a homomorphism of commutative rings with identity. Let $f_1, \ldots, f_n$ be elements of $A$, such that $(f_1, \ldots, f_n) = (1)$. I want to prove that if each of the ...
Adelhart's user avatar
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Question on the proof of localization of tensor product isomorphic to tensor product of localization

I have seen some posts here that talks about the isomorphism between the localization of a tensor product and the tensor product of localization, but I am not sure if I am understunding it. For ...
Superdivinidad's user avatar
1 vote
1 answer
52 views

Proof that quasi-isomorphisms form a localizing class in the homotopy category of complexes

I'm currently reading Gelfand and Manin's book Methods of Homological Algebra. Theorem III.4.4 says that the class of quasi-isomorphism in a homotopy category of complexes is localizing and I am ...
Albert's user avatar
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Show that $D(f) \subseteq \mathbb{A}_k^{n+1}$ is an affine algebraic set with coordinate ring $A[f^{-1}]$

I was trying to do the following exercise from: https://www.math.uni-bonn.de/people/ja/alggeoI/blatt01.pdf Let $X \subseteq \mathbb{A}^n(k)$ be an affine algebraic set with $D(X):=\{f\in k[X_1,\cdots,...
Ishigami's user avatar
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2 votes
2 answers
114 views

What is the stalk map for a morphism of affine schemes?

$\newcommand{\Spec}{\operatorname{Spec}} \newcommand{\O}{\mathscr{O}}$ Let $X=\Spec A$, and $Y=\Spec B$, and suppose that $f:X\rightarrow Y$ is a morphism coming from the ring homomorphism $\phi:B\...
Chris's user avatar
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Question on proof of Localisation Theorem in Tom Dieck's Transformation Groups (Thm III.3.8)

I have a question regarding the Localisation Theorem (III.3.8) in Tom Dieck's Transformation groups, which states the following: Let $G$ be a compact Lie group, $(X,A)$ be a finite-dimensional ...
Liva's user avatar
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Passing from global number field to local number field

I want to clarify my understanding of the localization of the global field. Let $K$ be a number field, that is a finite extension of $\mathbb Q$ or $\mathbb{F}_p(x)$. These are not complete fields. ...
MAS's user avatar
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3 votes
0 answers
55 views

Localization of locally compact commutative ring

If $A$ is a locally compact (Hausdorff) commutative ring and $S$ is a multiplicatively closed subset of $A$, is there any natural topology we can put on $A S^{-1}$ such that it also becomes a locally ...
Pedro Lourenço's user avatar
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Using Nakayama's lemma in non-local ring

Let $R$ be a Noetherian integral domain of dimension one and $\mathfrak{m}$ an ideal such that $\text{dim }\mathfrak{m}/\mathfrak{m}^2=1$ as an $R/\mathfrak{m}$-vector space. The localization of $R$ ...
Navid's user avatar
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Isomorphism locus of a morphism of objects in the derived category is Zariski open?

Let $R$ be a commutative Noetherian ring and $\text{Mod} R$ be the category of $R$-modules. Let $M,N\in \mathcal D(\text{Mod } R)$ with finitely generated homologies. Let $f: M\to N$ be a morphism in $...
Alex's user avatar
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1 answer
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Why natural map $M \rightarrow \prod M_{\mathfrak{p}}$ makes sense?

I am referring to this post where there is a natural map from $A-$module $M \rightarrow \prod M_p$ for $p$ are maximal ideals of $A$. I understand why the map is injective if it makes sense, but what ...
Mahammad Yusifov's user avatar
1 vote
1 answer
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Residue fields commute with quotients?

Is the following statement true? And if so, is my proof correct? Let $A$ be a ring and $q \subseteq p \subseteq A$ prime ideals. There exists a (unique) isomorphism $$ A_p/pA_p \xrightarrow{\cong} (A/...
Patrick Perras's user avatar
1 vote
1 answer
26 views

Localization: $(x) R_{\mathfrak{m}}=R_{\mathfrak{m}}=\mathfrak{a} R_{\mathfrak{m}}$ for $x\in \mathfrak{a}$ but $x\notin \mathfrak{m}$

I have a commutative ring $R.$ Let $\mathfrak{a}$ be a non-zero ideal in this ring and $\mathfrak{m}$ be a maximal ideal. Also, let $x$ be a non-zero element of $\mathfrak{a}$ such that $x\notin \...
Haldot's user avatar
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1 vote
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Show the formal power series ring is a faithfully flat algebra.

Suppose $S=P^{-1}F[x]$, the localization of $F[x]$ at $P$ where $F$ is a field and $P=(x)\backslash\{0\}$. Let $\hat{S}=F[[x]]$, the formal power series ring. Clearly there is a homomorphism $\Phi: S\...
Dick Grayson's user avatar
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1 vote
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The localization of the modul of homomorphisms

I want to be able to better understand the localization of homomorphisms $f:N \rightarrow M$ by considering them as the localization of $f$ in the ring $S^{-1}Hom(N,M)$. for that, I would be very nice ...
Aviv's user avatar
  • 105
2 votes
1 answer
39 views

Why is this localization thin?

I am studying this paper by Malkiewich and Ponto. I am unsure about one claim. Let $\Delta$ be the augmented simplex category. Denote by $\mathfrak{J}$ the wide subcategory of $\Delta$ consisting of ...
Learner's user avatar
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Clarifications on Smith Normal Form

I'm solving an exercise where I need to find the Smith normal form of a matrix. As I understood, what I need to do for a $2\times3$ matrix is to find the determinant of each of its $1\times1$ and $2\...
WittyCatchphrase's user avatar
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1 answer
84 views

Localization using a prime ideal

I'm approaching localization of rings using multiplicatively closed sets, and the obvious case of when we take the complementary of a prime ideal of a ring, that is always multiplicatively closed. ...
WittyCatchphrase's user avatar
2 votes
0 answers
46 views

Localization of functor category

Let $\Lambda$ and $\mathcal{C}$ be categories and $\mathcal{W}$ a subset of morphisms of $\mathcal C$ such that there exists the localization $L: \mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$. Denote ...
espacodual's user avatar
1 vote
1 answer
37 views

How to compute a minimal spanning set and the minimal spanning number of $\Bbb C[x,y]_{(x-1,y-1)}/(x^3-y^2)$?

Let $A=\frac{\mathbb C[X,Y]}{(X^3-Y^2)}$. I am asked to show that $\mathbf m=(\overline X-1,\overline Y-1)$ is a maximal ideal of $A$ which I have shown successfully. Now I am asked to compute the $\...
Kishalay Sarkar's user avatar
1 vote
1 answer
64 views

Chinese Remainder Theorem and ideals generated in localizations

In Milne's notes on algebraic number theory (https://www.jmilne.org/math/CourseNotes/ANT.pdf), on page 51, Corollary 3.14 and 3.15 both used the argument "use Chinese Remainder Theorem and look ...
spiderchips's user avatar
2 votes
0 answers
58 views

Diagrams in a model category

Let $\mathcal{M}$ be a model category with class of weak equivalences $W$ and $\mathcal{J}$ a small category. The category of diagrams $\operatorname{Fun}(\mathcal{J}, \mathcal{M})$ inherits a class ...
Brendan Murphy's user avatar
1 vote
2 answers
133 views

Let $R$ be a Noetherian commutative ring with zero nilradical and with any localization at a maximal ideal as a finite ring. Prove that $R$ is finite

Suppose that $A$ is a Noetherian commutative ring such that: (1) the nilradical (intersection of all prime ideals) of $A$ vanishes, and (2) localization at every maximal ideal is a finite ring. Prove ...
Squirrel-Power's user avatar
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0 answers
25 views

Why is it the case that $S^{-1}(\mathbb{Z}/(p_i^{e_i})) \cong \mathbb{Z}/(p_i^{e_i})$ for multiplicative set $S \subset \mathbb{Z}$ prime to $p_i$.

Let $S \subset \mathbb{Z}$ denote a multiplicative set, i.e. $1 \in S$ and if $a,b \in S$ then $ab \in S$. Let $p_i \in \mathbb{Z}$ be a prime, and let $e_i \in \mathbb{Z}^+$. Furthermore, let $S$ be ...
Ben123's user avatar
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0 votes
0 answers
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Free module over local ring $R$. [duplicate]

People often say that a module $M$ over a not necessarily neotherian local ring $R$ being projective is flat, and also free. However, some refer to the finitely-generatedness of $M$, i.e. $M$ being ...
Pierre MATSUMI's user avatar
0 votes
1 answer
61 views

An example of local and global containments in quaternion algebras

Let $p$ be a prime. Let $B_{p,\infty}$ be a (unique) quaternion algebra ramified at exactly at $p$ and $\infty$ with a standard basis $1,i,j, k=ij=-ji$. Let $K=\mathbb{Q}(i) \subseteq B_{p,\infty}$ be ...
Jason Dil's user avatar
  • 315
1 vote
0 answers
31 views

Forming modules of fractions is exact [duplicate]

Let $A$ be a commutative ring and $S\subseteq A$ a multiplicatively closed subset. Let $M$ be an $A$-module. We can form the rings/modules of fractions $S^{-1}A$ and $S^{-1}M$ respectively. See Atiyah ...
Sardines's user avatar
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0 answers
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How to calculate the center of the circumcircle of the triangle formed by three geographic coordinates

Where is the center of those 3 geographic coordinates as an example: ...
O. Durand's user avatar
  • 109
1 vote
1 answer
143 views

If $M$ is a finitely generated module over a local ring, then $M$ is free.

Given a local ring $A$ and a finitely generated projective module $M$, show that $M$ is also free. So I'm actually required to prove this question, although I believe that I managed to prove that $M$ ...
Ubik's user avatar
  • 488
2 votes
1 answer
145 views

Localization of $k[x]/(x^2)$ at $(x)$

I was wandering about the site, looking for interesting questions, and I stumbled upon an answer that I can't quite undertand. Unfortunately its creator isn't active on the site anymore so I cannot ...
kubo's user avatar
  • 2,067
1 vote
1 answer
88 views

Theorem 8, Section 3.4 of Hungerford’s Abstract Algebra

Let $S$ be a multiplicative subset of a commutative ring $R$ with identity and let $I$ be an ideal of $R$. Then $S^{-1}I= S^{-1}R$ if and only if $S\cap I\neq \emptyset$. Proof: If $s\in S\cap I$, ...
user264745's user avatar
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1 vote
1 answer
45 views

$X$ a scheme, $\mathfrak{p} \in \operatorname{Spec}(A) \subset X$. Then $Im(\phi: A\rightarrow A_{\mathfrak{p}})\subset D(f)$, $f\in A/\mathfrak{p}$

I’m trying to understand a step in this answer (all the other steps are pretty clear to me). I decided to post this as a new question instead of as a comment because the author has been inactive for ...
Gokimo's user avatar
  • 355
-2 votes
1 answer
93 views

How do I prove that this extension is integral? [closed]

Let $A$ be an integral domain, let $K$ be its field of fractions, and let $L$ be a finite extension of $K$. For a $\alpha \in L$, let $B=A[\alpha]$. Prove that exists a non-zero $a\in A$ such that the ...
Juan José Campos's user avatar
2 votes
1 answer
100 views

Sheaf on Zariski closed subset $Y$ is well defined

Let $(X,\mathcal{O}_X)$ be a scheme, and $Y$ a Zariski closed subset. For each $U\subset X$ open we defined the ideal of $\mathcal{O}_X(U)$, $I(U)$ to be: $$I(U)=\{s\in \mathcal{O}_X(U): \forall x\in ...
Chris's user avatar
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0 votes
0 answers
45 views

Is there a name for this presheaf?

I’m taking an introductory course on Scheme theory and I’m looking for a reference for the following construction that was covered in the course. Let $A$ be a commutative ring, $U$ an open subset of $\...
Gokimo's user avatar
  • 355
1 vote
1 answer
52 views

Verifying sheaf base axiom $2$ on $\operatorname{Spec} A$

Let $A$ be a commutative ring, and $\operatorname{Spec} A$ be the spectrum of $A$ with its Zariski topology. A base for this topology is given by distinguished opens $U_f$. I am trying to verify that ...
Chris's user avatar
  • 3,441
0 votes
1 answer
102 views

Defining the structure sheaf on an affine scheme

Let $A$ be a ring, then structure sheaf on $\text{Spec }A$ is given by a sheaf on the base of distinguished opens, $\mathcal{O}(U_f)=A_f$, where $A_f=f^{-1}A$, the localization of $A$ by the ...
Chris's user avatar
  • 3,441
0 votes
1 answer
79 views

How Should I Prove that the Localization of Quotients Commute from here?

Let S be a multiplicatively closed subset of R, M an R-module, and N a submodule of M. I want to prove that: S$^{-1}$(M/N) is isomorphic to (S$^{-1}$M)/(S$^{-1}$N). Attempt: Since $0$$\rightarrow$ N$\...
Mr Prof's user avatar
  • 451
0 votes
1 answer
64 views

Is Every Isomorphism on Tensor Product of Modules an R-module Isomorphism?

We have an R-module M. We also have S, a multiplicatively closed subset of R. I want to prove that there exists a unique R-module isomorphism f: S$^{-1}$R$⊗$M $\to$S$^{-1}$M, defined by f(${\frac rs}$⊗...
Mr Prof's user avatar
  • 451
2 votes
0 answers
49 views

Modules over a product of fields are locally free

Let $F$ be a field. I'd like to show that all $F\times F$-modules are locally free. I think I've done that, but I'm confused by some material I found about this. Could you please verify my proof? The ...
Object's user avatar
  • 339
0 votes
1 answer
68 views

Counterexample for non finitely generated module not annhilating

There's a common result that says for a commutative ring $A$ with a multiplicatively closed subset $S$ and a given finitely generated $A$-module $M$ we have $S^{-1}M=0 \iff sM=0$ for some $s \in S$. ...
Bhoris Dhanjal's user avatar

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