# 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.

567 questions
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### Show that $\equiv$ is a congruence on $M\times S$

I'm sorry if a similar question has been posted before, but I was unable to find one based on my searches. This is an extra practice problem for a number theory class. I've been trying to prove this ...
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### If $\mathfrak{p}$ is a prime such that $M_\mathfrak{p} \neq 0$, then $\mathfrak{p}$ contains an associated prime of $M$

I am studying from Serge Lang's Algebra (3rd edition), and in Chapter X Noetherian Rings and Modules, $\S2$ Associated Primes, we have the following proposition: Proposition 2.10. Let $A$ be ...
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### Localization of $k[t]$ at a prime ideal is not a finitely generated $k[t]$-algebra?

Let $k$ be a field (infinite and algebraicaly closed), consider the prime ideal $(t)$ in $k[t]$. Consider the localization of $k[t]$ at $(t)$: $k[t]_{(t)}$. Now $k[t]_{(t)}$ is a $k[t]$-algebra. I ...
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### Question about the ring $A_P$ of regular functions on a variety $V$

Note: I am doing an introductory course in Commutative Algebra and not Algebraic Geometry; This is just a quick application I am trying to understand. Also, rings are commutative with $1$. When ...
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### Valuation ring and localizations [closed]

Theorem. Let $D$ be an integral domain with identity. The following conditions are equivalent. (1) $D_P$ is a valuation ring for each proper prime $P$ in $D$. (2) $D_M$ is a valuation ring for each ...
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### Localisation of $\mathbb{Z}/(p^k)$.

I was looking at the wiki that explains localization. It says that the only way to localize $\mathbb{Z}/(p^k)$ is $\{0\}$. The argument is that the elements of $\mathbb{Z}/(p^k)$ are either units or ...
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### When does localisation behave badly?

Localisation seems to be a very useful tool in commutative algebra/number theory, and it seems like in every case I've come across, it behaves incredibly well. By behaves well, I mean that it is ...
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### Localization of finitely generated algebra

Let $R$ be a reduced finitely generated algebra over $\Bbb Z$. Let $T$ be a finite set of prime ideals of $R$. Let $S = \bigcap_{p \not\in T} R \setminus p$. 1) Is it true that $A := S^{-1}R$ is ...
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### Find maximal ideals in semi local ring with singular

I am trying to analyze the normalization $N$ of a local ring $A_{\mathfrak{m}}$ of a variety with Singular. The normalization (integral closure in its total ring of fractions) is semi-local with its ...
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### Isomorphism between localizations of graded ring $S_{(P)} \cong [S_{(f)}]_{PS_f \cap S_{(f)}}$

I know that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the prime ideals ...
I'm trying to prove that An element $\frac{a}{f^n}$ of $A_f$ is regular iff $\frac{a}{f^n}$ is regular in $A_p$, for all prime ideals $p$ of $A$ such that $f\notin p$. Where $A$ is a unitary ...