Questions tagged [localization]

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Localization and containment in commutative ring

Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...
Asad Albani's user avatar
4 votes
0 answers
94 views

A variation of the hammock localization

Let $W \subseteq \mathcal{C}$ be a wide subcategory of a category $\mathcal{C}$. The saturation $\overline{W}$ of $W$ is the class of maps of $\mathcal{C}$ which become isomorphisms in $\mathcal{C}[W^{...
Valery Isaev's user avatar
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Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?

Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...
hasManyStupidQuestions's user avatar
22 votes
1 answer
643 views

When does rationalization commute with homotopy fixed points?

Let $X$ be a $G$-space. There are a number of places in the literature where one can find the claim that under certain conditions rationalization and taking homotopy fixed points with respect to a ...
skupers's user avatar
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4 votes
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A question related to bousfield localization and nilpotent completion

I am reading Bousfield's paper entitled "The localization of spectra with respect to homology" (MSN). In that paper, Corollary 6.13 states that, if a ring spectrum $E$ has countable homotopy and ...
Surojit Ghosh's user avatar
5 votes
0 answers
132 views

Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)

$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
Bashar Saleh's user avatar
19 votes
1 answer
856 views

What is classified by generalised Eilenberg MacLane spaces?

Given an abelian group $A$, the Eilenberg MacLane spaces $K(A,n)$ represent the the nth cohomology group in $A$. In a similar vein, given an arbitrary group $G$ and a space $X$, maps to the ...
Patrick Elliott's user avatar
2 votes
1 answer
148 views

Special submodules over almost Dedekind domains

An integral domain $R$ is an almost Dedekind domain if for each maximal ideal $m$ of $R$, the ring $R_m$ is a Dedekind domain, where $R_m$ is the localization of $R$ at $m$. Question: Let $M$ ...
user140640's user avatar
3 votes
0 answers
264 views

Reduced Noetherian ring is the intersection of its localizations at primes associated to a nonzero-divisor

I shall quote proposition 11.3 of Eisenbud: Commutative algebra If $R$ is a reduced noetherian ring then an element $x\in K(R)$ belongs to $R$ if and only if the image of $x$ in $K(R)_P$ belongs to ...
user301120's user avatar
5 votes
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Embedding the Mészáros subdivision algebra in an Orlik-Terao localization

The following is an open question (Question 4.1) from my paper $t$-Unique Reductions for Mészáros's Subdivision Algebra (published version in SIGMA 2018, and slightly updated preprint version with ...
darij grinberg's user avatar
1 vote
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643 views

Completion of localization of completion

Let $(A,m)$ be a noetherian local ring, and let $p \subseteq A$ be a prime ideal. From this data, we can construct two rings: 1. We may localize $A$ at $p$, and then complete, obtaining the $pA_p$-...
Localization's user avatar
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Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?

Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...
Zhaoting Wei's user avatar
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Localizing prime ideals over Noetherian rings

Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ ...
user304582's user avatar
4 votes
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166 views

Do we have criteria of strict localization of a Grothendieck category?

Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...
Zhaoting Wei's user avatar
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4 votes
1 answer
269 views

Ore localization and model structures

The question is this: Suppose C is a category, with a given multiplicatively closed set of morphisms S ⊆ C. The role of the denominator conditions on S is rather similar to the role of a Quillen ...
Amnon Yekutieli's user avatar
1 vote
1 answer
223 views

modules whose every submodule is a homomorphic image

Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$. Can we characterize all ...
user521337's user avatar
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1 answer
295 views

When is this localization map injective, if at all?

Let $K$ be a number field and $E$ be an elliptic curve defined over $\mathbb{Q}$. Consider the localization map $$ E(K)\otimes \mathbb{Q}_p/ \mathbb{Z}_p \rightarrow \bigoplus_{v|p} E(K_v)\otimes \...
debanjana's user avatar
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19 votes
2 answers
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What is the correct definition of localisation of a category?

Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange. There appears to be a discrepancy in the literature regarding the ...
Luke's user avatar
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7 votes
2 answers
347 views

When a localization of a category is (non-)reflective?

Let $S$ be a class of maps of a category $\mathcal{C}$. The localization of $\mathcal{C}$ at $S$ is reflective when the localization functor $\mathcal{C} \to \mathcal{C}[S^{-1}]$ has a fully faithful ...
Valery Isaev's user avatar
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9 votes
4 answers
1k views

Localization of $\infty$-categories

In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such ...
Exit path's user avatar
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3 votes
0 answers
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Open Gromov-Witten invariants via lozalization with $\mathbb{C}^{*}$ (not $S^1$) action

Amplitudes of open A-model on a Calabi-Yau 3-fold $X$ with branes are given by the open Gromov-Witten invariants of $X$. It is known how to compute them if there is a toric action on a manifold, which ...
Andrey Feldman's user avatar
4 votes
0 answers
174 views

What kind of module is this?

Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{...
Ben Knudsen's user avatar
5 votes
1 answer
273 views

Factorization of Gabriel-Zisman localization construction?

My question concerns whether the Gabriel-Zisman localization construction $S^{-1}$ for categories admits a known factorization into a pair of commuting constructions $S^l$ and $S^r$. The localization ...
Ben Cooper's user avatar
10 votes
1 answer
647 views

Acyclic aspherical spaces with acyclic fundamental groups

A space $X$ (by which I mean a CW complex) is acyclic if its reduced singular homology $\tilde H_\ast(X;\Bbb Z)$ is trivial in all degrees. A discrete group $\pi$ is said to be acyclic if its ...
John Klein's user avatar
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3 votes
0 answers
318 views

Localization of the pushforward in equivariant cohomology

I am reading Nekrasov's paper and in page 2 he considers the $G \times T^2$ equivariant cohomology of the (compactified) moduli space $\tilde{M_k}$ of $U(N)$ instantons on $\mathbb{C}^2$. Here $G$ ...
Marion's user avatar
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1 vote
0 answers
41 views

When is genus the same as stable equivalence?

Suppose $M, N$ are two $R$-modules (I had in mind the group ring $R=\mathbb{Z}[G]$ for a finite group $G$). By localizing at a prime $p$ I mean $M_{(p)}\cong M\otimes_R R_{(p)}$. If $M$ and $N$ are ...
Sam Williams's user avatar
2 votes
1 answer
138 views

On maximal ideals of $k [X_i : i \in I ] $ where $k$ is a field , $I$ is an infinite set with $|k| > |I|$

Let $k$ be a field and $I$ be an infinite set such that $|k| > |I|$ . Let $R := k [X_i : i \in I ] $ and $m$ be a maximal ideal of $R$ ; then is it true that $m \cap k[X_i] \ne 0 , \forall i \in I$ ...
user avatar
1 vote
1 answer
118 views

Basic elements and localizations

Let $(R, \mathfrak{m})$ be a local domain and $x$ is a basic element of $\mathfrak{m}$, that is $x \in \mathfrak{m} \setminus \mathfrak{m}^2$. Let $P$ be a prime ideal containing $x$. Is it true that $...
Pham Hung Quy's user avatar
1 vote
1 answer
100 views

Functions on rings and polynomials with coefficients in a certain kind of localisation

Let $R$ be a commutative ring with unity and let $S$ be a multiplicatively closed subset of $R$ such that $S$ contains no zero divisor . So the canonical map $f : R \to S^{-1}R$ is invective , hence w....
user avatar
4 votes
1 answer
112 views

If $\{f\in R[x]\:|\:f\text{ monic}\}$ is a right denominator set, is $\{f^i\:|\:i\geq 0\}$ a right denominator set also?

Let $R$ be a right (and left) Noetherian ring and $T=R[x]$ its polynomial ring. It was shown by Stafford that the set $S=\{f\in T\:|\:f\text{ monic}\}$ is a right denominator set. So my question is, ...
Sam Williams's user avatar
6 votes
3 answers
986 views

Are localization functors always essentially surjective?

Let $\mathcal{C}$ be a category and $\mathcal{W} \subseteq \text{Arr}(\mathcal{C})$ a set (or class) of arrows. There are (at least) two notions of localization of $\mathcal{C}$ with respect to $\...
Nicolas Schmidt's user avatar
0 votes
1 answer
248 views

Analytic spread of localization of an ideal

Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$. Q) Are there ...
Cusp's user avatar
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4 votes
0 answers
208 views

Checking a monad is idempotent

I have a monad $T: \mathcal{C} \to \mathcal{C}$ on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection $G$ of compact, ...
Dylan Wilson's user avatar
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10 votes
1 answer
333 views

How to identify localization of categories?

Let $C, D$ be categories with finite limits, $F:C\to D$ be a essentially surjective functor that commutes with finite limits, and let $S$ be the set of morphisms of $C$ that become isomorphisms in $D$ ...
h__'s user avatar
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0 votes
1 answer
154 views

$R$ is $\mathbb{Z}$ graded ring and $0\neq f \in R_1,$ show that $R_f \cong S[X,X^{-1}]$ [closed]

Suppose $R$ is $\mathbb{Z}$ graded ring and $0\neq f \in R_1.$ Then I want to show that $R_f \cong S[X,X^{-1}],$ where $S=(R_f)_0$ and $X$ transcendental over $S.$ I wanted to use the isomorphism $...
Panja's user avatar
  • 111
0 votes
1 answer
176 views

Right localization of $R[x,x^{-1}]$ at monic $f\in R[x]$

Let $R$ be a right Noetherian ring and $S=\{f\in R[x]\;|\;f\text{ monic}\}$. It is a result of Stafford that $S$ is a right denominator set in $R[x]$, so in particular we can localize $R[x]$ at any $f\...
Sam Williams's user avatar
2 votes
1 answer
102 views

Localising a right Noetherian ring at a set of regular elements

Let $R$ be a right Noetherian ring, and $S$ a multiplicative set consisting of regular elements where $1\in S$ and $0\not\in S$. Does the right ring of fractions $RS^{-1}$ exist? This is what I know ...
Sam Williams's user avatar
3 votes
1 answer
228 views

Localization of the pullback diagram

In the paper, Topologically Defined Classes of Commutative Rings, localization of the pullback diagram (with $v,$ surjective) $$ \begin{array} DD & \stackrel{v\ '}{\longrightarrow} & A \\ \...
user 1's user avatar
  • 1,345
2 votes
0 answers
494 views

Irreducibility over the field of fractions of a quotient of a polynomial ring

Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_0, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...
Nils Amend's user avatar
3 votes
2 answers
582 views

Localization of a symmetric monoidal category is monoidal when the morphisms being inverted are closed under tensor product

In the answer to question Localization of symmetric monoidal category, it was mentioned that '' Assuming that the tensor product of two morphisms in $S$ is again in $S$, the localised category should ...
kousaka's user avatar
  • 131
6 votes
1 answer
307 views

Localization, Slice Tower, and Motivic Spectra

Suppose $k$ is an algebraically closed field of characteristic $p>0$. There is an $\infty$-category of motivic spectra over $k$, denoted $\mathcal{S}pt(k)$. As in algebraic topology, there are ...
pj12's user avatar
  • 63
0 votes
0 answers
86 views

When does an automorphism extend to a localisation (noncommutative rings)

Let $R$ be a (not necessarily commutative) ring. Let $\tau$ be an automorphism of $R$. Consider the localisation of $R$ at a set of multiplicative elements which satisfy the ore condition, say $X$. ...
No1729's user avatar
  • 201
1 vote
1 answer
167 views

Colocal Objects in Enriched Bousfield Colocalizations

Let $C$ be a $V$-model category, and $\mathcal{K}$ a set of objects of $C$. Let me denote (derived) simplicial homotopy function complexes by $\text{Dmap}$ and derived $V$-function complexes by $\text{...
Alexander Körschgen's user avatar
6 votes
0 answers
1k views

Verdier Quotient a quotient?

This question seems trivial, so hopefully it will be resolved quickly. As pointed out in this question on quotient categories and localization, the two constructions are sometimes related, but in ...
Spencer Leslie's user avatar
30 votes
1 answer
755 views

Is a filtered colimit of rational spaces again rational?

Let me first explain the statement of the question and then give some indication why the answer might be 'yes'. By a space I mean, say, a simplicial set and by rational I mean rational in the sense of ...
Thomas Nikolaus's user avatar
13 votes
3 answers
895 views

Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products

A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the reflector. If the category $C$ is ...
Sergei Ivanov's user avatar
4 votes
0 answers
418 views

Localization in equivariant cohomology theory for groups other than ($p$-)tori

Recall the following localization theorem, as stated in Hsiang's Cohomology Theory of Compact Transformation Groups: Theorem. Let $G=(S^1)^k$ be a torus, $X$ a paracompact $G$-space with finite ...
user3158840's user avatar
7 votes
0 answers
251 views

Topological localization of (infinite) inverse limits

The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological ...
Tyrone's user avatar
  • 5,016
4 votes
1 answer
865 views

Using and understanding the Atiyah-Bott localization theorem/integration formula

I posted this on r/math, but was told I might have better success here given the level of the question. Basically, I need to learn how to use the localization theorem to compute integrals on ...
PointlessGraph's user avatar
7 votes
1 answer
320 views

Is an abelian category a Serre subcategory of its ind-category?

Let $\mathcal C$ be an abelian category and consider its ind-category $Ind(\mathcal C)$: (1) Is $Ind(\mathcal C)$ always abelian? (If not, what conditions are needed?) (2) Is $\mathcal C\subseteq ...
gocardinals's user avatar