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\...
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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 ...
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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}}...
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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 ...
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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,\...
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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 ...
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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 ...
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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 $...
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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{...
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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 ...
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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 ...
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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 ...
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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,...
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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\...
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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 ...
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