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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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Is $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ an integral domain and is $(A/\mathfrak{q})_\mathfrak{p}$ a ring?

Let $A$ be a commutative ring with identity and $\mathfrak{q}\subset\mathfrak{p}$ two prime ideals of $A$. I am trying to determine whether $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ is an integral ...
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Localization at annihilators of an ideal

I was reading this post and on line +10-11 of the proof of lemma 27.25.1, it seems to claim the following: Let $A$ be a ring, $I \subseteq A$ an ideal, and $M$ an $A$-module. Let $M_I:=\{ x \in M\...
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Support of localization of a module at a minimal prime over the support of the original module

If $M$ is a non-zero module over a commutative ring $R$ (not necessarily Noetherian), and $P$ is a minimal prime in $\mathrm{Supp}(M)$, then is it true that $\mathrm{Supp}(M_P)=\{PR_P\}$ (where $M_P$ ...
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The form of prime ideals in $S^{-1}R$

Theorem. Let $R$ be a commutative ring with $1_R$, $P\in \mathrm{Spec}(R)$ a prime ideal of $R$, $S\subseteq R$ an multiplicative subset of $R$ and $\nu:R\longrightarrow S^{-1}R,\ a \longmapsto \...
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1answer
31 views

Equivalence for rings with localization property

I feel like I need a hint for the following exercise: Let $R$ be some commutative unitary ring. If $M$ is a $R$-module, let $M[f^{-1}]$ denote the localization of $M$ with respect to the set $\{ f^...
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Broad question on morphisms of stalks of quasi-coherent sheaves on schemes

This question was inspired by reading about a criterion for a morphism into projective space (over an algebraically closed field) to be a closed immersion based on local rings. It got me thinking ...
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Finitely generated projective modules, over commutative Noetherian ring, of constant rank $1$, if stably isomorphic, then isomorphic?

Let $M,N$ be finitely generated projective modules over a commutative Noetherian ring $R$ such that $M_P \cong N_P \cong R_P, \forall P \in Spec(R)$. If $\exists n\ge 1$ such that $M \oplus R^n \...
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The elements of the extended ideal $I^e\trianglelefteq S^{-1}R$

Let $R$ be a commutative ring with $1_R$, $S\subseteq R$ a multiplicative set, $I\trianglelefteq R$ and ideal of $R$ and $\nu:R\longrightarrow S^{-1}R,\ a\longmapsto \nu(a):=\frac{a}{1_R}$ the natural ...
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Localization of Smooth Functions on R

Consider the ring of functions $C^{k}(\mathbb{R}^d;\mathbb{R}^d)$ from $\mathbb{R}^d$ to itself with $k$-continuous derivatives. Is the localization of $C^{k}(\mathbb{R}^d;\mathbb{R}^d)$ at the ideal ...
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Co-augmented functor and equality of natural transformations (and idempotent monads)

Suppose we have a category $C$ together with an endofunctor $L:C\to C$ and a natural transformation $\eta : Id_C\to L$. Suppose furthermore that the two natural transformations $L\eta, \eta L :L\to LL$...
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Show that this mapping between localized modules is an isomorphism

Let $R$ be a ring. Let $M$ and $N$ be $R$-modules where $M$ is finitely presented. Then for every multliplicative set $S \subset R$ the canonical mapping $~~~~~~~~~~~~~~~~~~~~~~~~~~~~Hom_R(M,N) \...
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Does $S^{-1}I \subset S^{-1}J$ imply $I \subset J$?

Let $S$ be a multiplicative subset of a commutative ring with identity, and consider the ring of fractions $S^{-1}R$. Ideals in $S^{-1}R$ of are of the form $S^{-1}I$, where $I$ is an ideal in $R$. ...
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How does $g \notin \mathfrak p \implies$ $g$ is nowhere vanishing on $V$?

I am studying localization right now from the lecture notes given by our instructor. In this notes he describes an example of localization what he told us yesterday in the class. Here's this $:$ Let $...
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A characterization of principal rings

I would like to know if the following characterization (for principal rings, not necessarily domains) is true: $ A $ is principal $ \leftrightarrow $ $ A_\mathscr{M} $ is principal $ \forall \mathscr{...
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Find example to show that the containment $\text{Supp}(M \otimes_R N) \subset \text{Supp}\, M \cap\text{Supp}\, N$ is proper [duplicate]

$$(M \otimes_R N)_P\cong(R-P)^{-1}(M \otimes_R N)\cong(R-P)^{-1}M \otimes_R (R-P)^{-1}N\cong M_P \otimes_R N_P$$ This implies $\text{Supp}(M \otimes_R N) \subset\text{Supp} M\, \cap \text{Supp}\,N$. ...
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1answer
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Isomorphism of modules localized at a maximal ideal

I'm currently trying to understand Quillen's original proof of the Quillen-Suslin theorem on projective modules over polynomial rings and I'm having difficulty justifying a claim in the paper. On the ...
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Why does $Re=S^{-1}R$?

The problem is to show that if $e$ is an idempotent in a ring $R$, then $Re=S^{-1}R$ where $S=\{1,e,e^2,e^3,\dots\}=\{1,e\}$. In fact this doesn't even seem plausible to me, because $Re$ is "smaller" ...
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Locally Noetherian Domain With Finitely Many Prime Ideals

Let R be a domain with finitely many prime ideals such that the localization at each prime, $R_{\mathfrak p}$, is Noetherian. Then is $R$ necessarily Noetherian?
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Non-Noetherian Domain Which is Locally Noetherian

Let $R$ be an integral domain such that the localization, $R_{\mathfrak p}$, at each prime ideal, $\mathfrak p \le R$ is Noetherian. Then is $R$ necessarily Noetherian? In the case of $R$ not ...
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1answer
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Injectivity between two localizations

Let $R$ be a commutative ring and $\mathfrak{p}\subseteq\mathfrak{q}$ be two prime ideals. We send $R_\mathfrak{q}\to R_\mathfrak{p},r/s\mapsto r/s$. Is this map always injective? I was considering ...
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On natural homomorphism $\nu:R\longrightarrow S^{-1}R$

Let $R$ be a commutative ring with $1_R$ and $S$ an multiplicatively closed set. We define the natural homomorphism \begin{align*} \nu:R &\longrightarrow S^{-1}R, \\ a&\longmapsto \nu(a):=\...
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Localization of a ring vs. localization of a module

Let $A,B$ be (noetherian) domains and consider $\phi: A \rightarrow B$ a morphism of rings. This map endows $B$ with an $A$-algebra structure. Consider the induced map $\psi: \mathrm{Spec}\,B \...
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Ideal with zero localizations at prime ideals containing it

Let $R$ be a commutative unital ring. I know that if an $R$-module has zero localizations at all prime ideals of $R$, then it is a zero module. Consider a proper ideal $I\subset R$ as an $R$-...
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If $Q \in\mathrm{Spec}(B)$ is the unique prime ideal lying over $P \in\mathrm{Spec}(A)$, then $B_P=B_Q$.

Let $A \subseteq B$ be an integral extension of integral domains. Suppose $P \in \mathrm{Spec}(A)$ and $Q \in \mathrm{Spec}(B)$ is the unique prime ideal lying above $P$. Prove that $B_Q=B_P$. My ...
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Show that if a submonoid $S$ of a commutative monoid $M$ contains an absorbing element, then the localization of $M$ by $S$ contains only one element.

Here is the full question: If $M$ is a commutative monoid, $S\subset M$ is a submonoid and there is a $z\in S$ that is absorbing in $M$ (i.e. $zm=z$ for all $m\in M$), then show that $M_S$ has only ...
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1answer
38 views

Prime Ideal In Left-Localization

Let $A$ be a left and right-Noetherian ring and let $S \subset A$ be left-localisable. Let $P \subset A$ be a prime ideal such that $P \cap S = \emptyset$. I wish to show $S^{-1}AP = \{\phi(s)^{-...
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Analogue of Krull intersection theorem for Symbolic powers of ideals

All rings below are commutative with unity and Noetherian. Let $R$ be a domain or a local ring and $J$ be a proper ideal. Is it true that $\bigcap_{n>1} J^{(n)}=(0)$ ? If this is not true under ...
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Two Sided Ideal in Left-Localization of a Left-Noetherian Ring

Let $R$ be a left-Noetherian ring, and suppose $S \subset R$ is left-localizable ($S$ is assumed to be a multiplicative closed subset and $1 \in S$). Let $\phi: R \to S^{-1}R$ be the canonical ...
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Rings of fractions

If $R$ is a ring, $I$ is an ideal of $R$ such that $I\ne R$, and $S$ a multiplicatively closed set of $R$, and we have that this property is true, $$\frac{a}{s} \in S^{-1}I \implies a\in I$$ Can we ...
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When is this statement true $\frac{a}{s} \in S^{-1}I \implies a \in I$?

If $R$ is a ring, $I$ an ideal of $R$, and $S$ a multiplicatively closed subset of $R$. We know that if $a \in I$ then $\frac{a}{s} \in S^{-1}I$. But the converse isn't true always. If $\frac{a}{s} \...
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1answer
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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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1answer
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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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1answer
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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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Commutative ring with finitely many minimal primes [duplicate]

$A$ be a commutative ring with $1$ with finitely many minimal primes $\{p_1,\ldots,p_n\}.$ Then how can I show that $S^{-1}A \cong A_{p_1} \times\cdots \times A_{p_n},$ where $S=A \setminus \bigcup_{i=...
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Intersection of the kernels of localization maps

Let $M$ be a finitely generated module over a Noetherian ring $R$. I need to show that for a multiplicately closed subset $U\subset R$, $$\bigcap_{P\in \operatorname{Ass}(M)\\ P\cap U=\emptyset}\ker(M\...
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2answers
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if $M \otimes_A (A_m/mA_m)=0$ for every maximal ideal $m \subset A$, then $M=0$, $M$ finitely generated

Suppose $M$ is a finitely generated $A$-module. Prove that if $M \otimes_A (A_m/mA_m)=0$ for every maximal ideal $m \subset A$, then $M=0$. Subscrpit $_m$ means localization at $m$. First consider ...
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1answer
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The localization of the ring $\mathbb{Z} \times \mathbb{Z}$ at every prime ideal is an integral domain

I want to show that the localization of the ring $\mathbb{Z} \times \mathbb{Z}$ at every prime ideal is an integral domain. $\mathbb{Z} \times \mathbb{Z}$ is not an integral domain since $(0,1)\cdot(...
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1answer
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The map $M[f^{-1}]\to M[f^{-1}g^{-1}]$

Let $M$ be a module over $R$. When one talks about the map $M[f^{-1}]\to M[f^{-1}g^{-1}]$, which map do they mean? My guess is that $M[f^{-1}][g^{-1}] \simeq M[f^{-1}g^{-1}]$, and then the above map ...
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Localization of hereditary rings

A (left) hereditary ring $R$ is a ring for which all submodules of projective (left) $R$-modules are again projective. If $R$ is commutative, and $S\subseteq R$, then $S^{-1}R$ is hereditary :https://...
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$R_{\mathfrak{p}}$ is integrally closed in its total quotient ring, then $R$ is integrally closed in its total quotient ring.

It is known that if $R$ is a Noetherian ring, then an element $x\in K(R)$ in its total quotient ring belongs to $R$ iff the image of $x$ in $K(R)_{\mathfrak{p}}$ belongs to $R_{\mathfrak{p}}$ for ...
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On modules in which tensor of each two elements is commutative

Let $R$ be a commutative ring with unity. Let $M $ be a finitely generated $R$-module. Is it true that the following conditions are equivalent : (1) In $M \otimes_RM$, we have $m\otimes n=n\otimes m,...
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1answer
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An equivalent condition for the localization map $M\to M_P$ to be injective

How do I show that given a prime ideal $P\subset R$, the map of $M\to M_P, m\mapsto m/1$ is injective if and only if $R\setminus P$ contains no elements killing a nonzero element of $M$? One ...
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Show that every non-zero ideal of the ring of rationals with odd denominator is generated by $2^{n}$ [duplicate]

Let $R \subset \mathbb{Q}$ be the subring $\left\{\frac{a}{b} \mid a, b \in \mathbb{Z}, b \text { odd }\right\}$. Prove that the ideals of $R$ are the zero ideal $\{0\}$ and $2^{n} R$ for $n \geq 0$. ...
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$K[X]_X$ is not integral over $K[X]$

The localization $K[X]_{X}$ is a ring extension of $K[X].$ I want to show that $K[X]_X$ is not integral over $K[X]$ using lying above. I tried to find a maximal ideal in $K[X]_X$ whose contraction in ...
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Is it possible to localise the renormalisation group?

The renormalisation group allows you to consider what happens to a model when considered on different spatial scales. It is not a group, reflecting the fact that you can always move to a larger ...
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1answer
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Do the maps of rings coincide under the condition of reducedness and localization assumption?

Say $f,g:B\rightarrow A$ are maps of rings such that $A$ is reduced and $i_{\mathfrak p}\circ f=i_{\mathfrak p}\circ g $ for all primes $\mathfrak p\subset A$ where $i_{\mathfrak p}:A\rightarrow A_{\...
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1answer
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Localization of the polynomial ring at a prime ideal modulo maximal ideal is isomorphic to polynomial ring modulo prime ideal.

Let $p \in K[T]$ irreducible, s.t. $\text{LC}(p) = 1$. Then $$ K[T]/(p) \cong K[T]_{(p)}/pK[T]_{(p)}.$$ What I have is: \begin{align*} &K[T] \hookrightarrow K[T]_{(p)} \text{ and } K[T]_{(p)} \...
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1answer
56 views

Ring of fractions of $\mathbb Z_6$

I am studying commutative algebra and I have a problem in calculating the fraction rings. For example, it would be a great help someone help me to calculate the ring of fraction if $ S= \{1,3,5\} $ ...
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1answer
29 views

Is $\mathbb{C}[[t]]$ isomorphic to the localization of $\mathbb{C}[t]$ at the maximal ideal $(t)$?

It seems to me that the power series ring $\mathbb{C}[[t]]$ is isomorphic to the localization of $\mathbb{C}[t]$ at the maximal ideal $(t)$, but I am not sure.
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1answer
85 views

Infinite direct sum of p-adic integers is not p-adic

Studying Bousfield localization I stumbled upon this elementary example: we start with $\mathcal{D}$ the derived category of $p$-local abelian groups and we can consider the Bousfield class of $\...