# 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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### Is being finitely generated a local property

Searching on this site and others leads to lots of dicussion about localisation at multiplicatively closed subsets of the form $\{f_i^j\}_{j=1}^\infty$ where $\{f_i\}_{i=1}^n$ generate the whole ring ...
1 vote
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### Krull dimension of the local ring at the generic point of a divisor is 1.

Let $X$ be a nonetherian integral separated scheme which is regular in codimension one, i.e. every local ring $\mathscr{O}_x$ of $X$ of dimension one is regular. Let $Y$ be a prime divisor, i.e. a ...
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### Why does $1/f$ count as a regular polynomial function on $D(f)$?

In elementary treatments of algebraic varieties, a regular morphism between affine varieties is one whose components are all polynomials, and the coordinate ring as a ring of polynomial functions. ...
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### Nonzero elements in localization

I have a very basic question that's been vexing me. Let $a\in A$ be a nonzero element of a Noetherian ring. Then $a$ lies in some maximal ideal $M$. Consider the localization $A_M$ at that maximal ...
1 vote
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### Is every local martingale right continuous?

Is every local martingale right càdlàg (i.e. right continuous with left limits)? At the university, in the definition of martingale we assume martingales to be right càdlàg processes. We call an $X$ ...
1 vote
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### Functor with two successive adjoints is a localization

Assume that we have a fully faithful functor $f_{!} \colon \mathcal{C} \to T^{-1}\mathcal{P}(\mathcal{G})$ where $\mathcal{G}$ is some small $\infty$-category and $T$ is a strongly saturated class of ...
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### Multiplicity of intersection of $y^2=x^3$ and $x^2=y^3$ at the origin

Those curves intersect at the origin with multiplicity 4, if I did everything correctly. In fact, parametrizing by $t \mapsto (t^2,t^3)$ the first curve and plugging into the second, yields $t^4=t^9$, ...
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### A normal ring (possibly a non-domain) is integrally closed in its total ring of fractions

$\newcommand\frp{\mathfrak{p}}$I am trying to understand 034M of the Stacks Project, whose statement is the title of this post. The proof seems to implicitly assume that $R$ is a subring of $R_\frp$ (...
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### Localising a ring twice gives ring isomorphic to localising by each subset in turn

Given a ring $R$, suppose we have two multiplicative subsets $S,U \subseteq R$ (which contain 1). Write $US = \{us \mid u \in U, s \in S\}$ also a multiplicative subset, containing both $S$ and $U$ ...
1 vote
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### Definition of $k$-rational points on an algebraic set

I am learning introductory algebraic geometry by myself. Probably I am misunderstanding something. Could you point out where I mistake? Let $k$ be a field that is not necessarily an algebraically ...
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### How to derive this from the universal property of the ring of fractions?

(All the involved rings are commutative with identity.) The "universal" definition that I follow is this: Let $S$ be a multiplicative subset of a ring $A$ which also contains $1_A$. Then a ...
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### Consider the ring $A = \mathbb{Z}/60\mathbb{Z}$ and the prime ideal $\mathfrak{p} = (2)$. Describe the localization $A_{\mathfrak{p}}.$

Consider the ring $A = \mathbb{Z}/60\mathbb{Z}$ and the prime ideal $\mathfrak{p} = (2)$. Describe the localization $A_{\mathfrak{p}}.$ Apparrently this should be $\mathbb{Z}/4\mathbb{Z}$, but I don'...
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### germ at origin: plane nocal cubic and x,y-axis

I am reading I.5 of Hartshorne's AG. In the example I.5.6.3, he explains how "completion" helps with analyzing the local structure of varieties by demonstrating two examples of completion: ...
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### Let $M_1, \dots, M_n$ be $A$-modules. Describe an isomorphism $S^{-1}(M_1 \times \dots \times M_n) \to S^{-1}(M_1) \times \dots \times S^{-1}(M_n).$

Let $M_1, \dots, M_n$ be $A$-modules. Describe an isomorphism $$S^{-1}(M_1 \times \dots \times M_n) \to S^{-1}(M_1) \times \dots \times S^{-1}(M_n).$$ This is essentially showing that the ...
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### Localization of module finite extension of regular ring

Let $R\subseteq S$ be an extension of Commutative Noetherian rings such that $R$ is a regular ring and $S$ is module finite over $R$. Let $P$ be a prime ideal of $S$, then $P\cap R$ is a prime ideal ...
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### Commuting operators vs. commutative rings, and localization in the quantum mechanical sense vs. commutative algebra sense.

One of the answers in this MO post https://mathoverflow.net/questions/7917/non-commutative-algebraic-geometry draws a connection between localization in commutative algebra (or the failure thereof), ...
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### An important proposition that relies on the group of fractional ideals of a Dedekind domain and localization

Does the map from the group of fractional ideals of a Dedekind domain $A$ to the group of fractional ideals of $S^{-1}A$ where $S$ is a multiplicative subset is surjective? And what about its kernel? ...
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### In Lemma 10.53.5 of Stacks (about commutative Artinian rings), how did they use localisation?

Link here: https://stacks.math.columbia.edu/tag/00J4#:~:text=A%20ring%20R%20is%20Artinian%20if%20and%20only%20if%20it,localizations%20at%20its%20maximal%20ideals Lemma 10.53.5. Any ring with finitely ...
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### Dimension of $(A[[X]])_\mathfrak{m} \geq k+1$, regular ring, chain of prime ideals

Assume $A$ is a regular ring and $m$ a maximal ideal of $A$. We define $R= A[[X]]$.Then $\mathfrak{M}=mR + XR$ is a maximal ideal of $R$. Lets assume $h(mA_m) = k$. I want to show, that \begin{align*} ...
1 vote
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### Localization in power series

Let $A$ be a comm. ring with unity. Say $\mathfrak{M}$ is a maximal ideal of $A[[X]]$. Is following statement generally true? \begin{align*} (A[[X]])_\mathfrak{M} \cong (A_{\mathfrak{M} \cap A}[[X]])_\...
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### General method of finding dimension of a ring and determining regularity

Let $R=\mathbb{Z}[x,y]$, $A=R/(y^3-x^3-4)$ and $m=(x,y,2)$. Now I want to find out what the dimension is of $A_m$ and say whether it is regular or not. My prefered definition of the ring $A_m$ being ...
1 vote
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### Prove that the following localization is a local PID

Consider a quadratic field $\mathbb{Q}\sqrt{-15}$ and $R$ as its algebraic integers, let $\mathfrak{p}$ = $(2,\frac{1+\sqrt{-15}}{2})$, then it is rather not difficult to show $\mathfrak{p}$ is a ...
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### Applications of a localization of a ring other than algebraic geometry

The localization construction is extremely useful in algebraic geometry. But this object seems for me very natural (of course, that's maybe only a mistake of my immature mind) for commutative rings ...
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### Query in contraction of localization

I was reading through Glaz's Commutative Coherent ring book, there I encountered a theorem where a part of it stated that For two commutative rings with unity $S$ and $R$, where $S$ is a $R-$ algebra ...
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### Inclusion of Rings after Localization

Let $\phi:A \to B$ an injective finite ring map between noetherian integral domains $A,B$. Let $C \subset B$ a subring of $B$ and assume that there exist a prime ideal $\mathfrak{p} \subset A$ , ...
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### Equivalent definitions of Saturated multiplicative system in a category

$\DeclareMathOperator{\id}{id}$I'm discovering the localization of categories, subcategories and functors. Amongst the first thing that comes up in this chapter, in the book I'm reading, is the ...
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### Is the stalk of an irreducible scheme at the generic point always a field?

Let $X$ be an irreducible scheme. It can be proved that it has a unique generic point, i.e. there is a unique point $\xi \in X$ such that $\overline{\{ \xi \}} = X$. One can identify $\xi$ as the ...
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### Noetherian localization of a commutative ring

I realize this question has been answered multiple times (here for example), but I still can't figure out this problem even after looking at every solution given. The problem statement is: Prove that ...
I'm reading this answer, in which, one of the step involves showing that: $$s_3(r_1s_2-r_2s_1)=0,\quad s_1(r_2s_3-r_3s_2)=0\implies s_2(r_1s_3-r_3s_1)=0$$ I am utterly confused on how this implication ...