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 ...
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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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Localizing a ring twice
Let $R$ be a ring and let $S \subset R$ be a multiplicative system, i.e. for all $f,g \in S$ it follows $fg \in S$. Let $S^{-1}R$ denote the localization of $R$ that is defined using the relation on $...
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If $M$ and $N$ are finitely generated $R$-modules such that their tensor product is $0$, then their annihilators over $R$ are comaximal
I am trying to prove this using the local-global principle. Let $I = Ann_R(M) + Ann_R(N)$.
It suffices to show that $I_\mathfrak{m} = R_\mathfrak{m}$ for all maximal ideals $\mathfrak{m}$ containing $...
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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 ...
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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 $X$ ...
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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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Why $S^{-1}R$ is called localization, despite $S^{-1}R$ is not always local ring?
Let $R$ be unital commutative ring, and $S\subset R\setminus\{0\}$ be multilplicative set. Then $S^{-1}R=(R\times S)/\sim$ where $(a,s)\sim(b,t)\Longleftrightarrow \exists u\in S\text{ s.t. }u(ta-bs)=...
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$B/\mathfrak p B\cong B_\mathfrak p/\mathfrak p B_\mathfrak p$ for Dedekind extensions $A\subseteq B$ and primes $(0)\ne\mathfrak p\subset A$
I'm studying number theory at the moment and came across a step in a proof I cannot resolve. The setting is the following:
Let $A \subseteq B$ a finite ring extension of Dedekind domains, i.e. one-...
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Ring of formal multivariate Laurent series as a consecutive localisation
How do we localise a the ring of formal power series $\mathbb C[[x, y]]$ to get the field of multivariate Laurent series $\mathbb C((x))((y))$? If $S_x$ and $S_y$ are the multiplicative closed sets $\{...
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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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Scheme theoretic fiber vs. Degree of a point of a finite morphism of affine schemes
Let $f: R\to S$ be a finite morphism of Commutative Noetherian rings, so $S$ is module finite over $R$ via $f$. Let $Q \in \text{Spec}(S)$, and set $P=f^{-1}(Q)$, so we have an induced map $R_P \to ...
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Dimension of a positively graded ring after a suitable localization
Quesion- Let $R=\bigoplus_{i\ge 0} R_i$ be a (non-trivial) positively graded commutative Noetherian ring with $1(\not=0)$ of (Krull) dimension $d\ge 0$. Let $S\subset R_0$ be a multiplicative set such ...
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Is testing whether the nonunits of $K[x,y]_{(x,y)}[1/y]$ form an ideal a messy undertaking or am I going about it wrong?
I'm studying the ideals of $R = K[x,y]_{(x,y)}[1/y]$: the polynomials in $1/y$ with coefficients in the localization of $K[x,y]$ at the ideal $(x,y)$. I've managed to identify that its units are given ...
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Is the localization of a polynomial ring at a prime ideal a valuation ring if and only if the prime ideal is principal?
Is the localization of a polynomial ring at a prime ideal a valuation ring if and only if the prime ideal is principal?
It seems to me that it is the case, and here's my work on the question in the ...
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Is the localization at a prime ideal of any polynomial ring always a valuation ring?
I observed that all examples of localizations of polynomial rings at prime ideals I've encountered have been, so far, valuation rings, and so I started wondering whether this is true in all cases. I ...
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Sheaf topos is localisation at covering sieve inclusion
I know that the sheafification functor $a:Pr(C) \to Sh(C,J)$ is up to equivalence the localisation $Pr(C) \to Pr(C)[W^{-1}]$ at the class of those morphisms which $a$ inverts. But is it also the ...
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stalk isomorphism + homeomorphism = isomorphism of varieties
I'm going over some old exercises in Chapter I of Hartshorne and am stuck on the reverse direction of Exercise I.3.3(b), which should follow straight from definitions. If $\varphi:X\to Y$ is a ...
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Intuitively, why $\operatorname{spec}(S^{-1}A) \cong \lbrace \mathfrak{p} \in \operatorname{spec} (A)| \mathfrak{p} \cap S = \emptyset \rbrace$?
If $A$ is a commutative ring, $S$ is a multiplicative subset of $A$, I’d like to understand intuitively why is there a bijection between $\operatorname{spec}(S^{-1}A)$ and $\lbrace \mathfrak{p} \in \...
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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 $U$
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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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An exercise from Fulton related to localization.
Need a little help with the following problem.This problem appears in Fulton's book on algebraic curves.The question is as follows:
Let $p=(0,0,...,0)\in \mathbb A^n$ and $\mathcal O_p=\mathcal O_p(\...
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Extension and contraction of prime ideals by ring homomorphism
Let $A$ and $B$ be commutative rings with $1 \neq 0$. Let $\varphi$ be a ring homomorphism with $\varphi(1) = 1$. We consider the extension and contraction of $\varphi$. Let $P \subset A$ be a prime ...
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Enveloping Algebra and localization on maximal ideal.
Let $k$ be a field and $A$ a commutative $k$-algebra. The enveloping algebra of $A$, denoted by $A^e$ is defined as $A\otimes_kA^o$ where $A^o$ is the opposite algebra. Here the opposite algebra does ...
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Why is it that $(\mathbb{C}[x]/(x^2-x))_{x-1} \cong \mathbb{C}?$
Consider $\mathbb{C}[x]/(x^2-x)$. Why do we have for the localization at $x-1$ that $$(\mathbb{C}[x]/(x^2-x))_{x-1} \cong \mathbb{C}?$$
We are essentially inverting at $\{1, x-1, (x-1)^2, (x-1)^3, \...
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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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Assume $M^{\tau_n}$ is uniformly integrable. Is $X^{\tau_n}$ uniformly integrable?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a filtration. Let $M$ be a real-valued continuous local martingale w.r.t. $\mathcal G$. Let $(\...
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Homogeneous prime ideals in $(S_\star)_f$ map bijectively to homogeneous prime ideals in $(S_\star)$ not containing $(1,f,f^2,...)$?
As the title suggests, why do homogeneous prime ideals in $(S_{\star})_{f}$ map bijectively to homogeneous prime ideals in $S_{\star}$ not containing $(1,f,f^2,...)$?
What I tried: I know that prime ...
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How to show $(k[x,y,z]/(xz,yz))_z\cong k[z]_z$?
As the title describes, how to show that $(k[x,y,z]/(xz,yz))_z$, $k[x,y,z]/(xz,yz)$ localized at $(1,z,z^2,...)$, is isomorphic to $k[z]_z$?
I have tried using a theorem saying that if the sequence $M'...
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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*}
...
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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 ...
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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 ...
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Showing that equivalence in ring of fractions is well defined [duplicate]
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 ...
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On the right cancellation in localization of category. Is it necessary?
I recently faced the concept of derived category, introduced as localization of homotopy category. I tried to verify of the axioms and I stucked in the following: In the definition of multiplicative ...
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