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4
votes
1answer
143 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 \...
10
votes
1answer
492 views

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 ...
7
votes
2answers
172 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 ...
7
votes
4answers
376 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 ...
5
votes
0answers
37 views

Do twisted group rings of free abelian groups admit universal fields of fractions?

Let $R$ be an associative ring with unit. Recall that an epic $R$-field is a ring epimorphism $\alpha\colon R\to D$ to a skew field/division ring $D$. An epic $R$-field $\alpha$ is called a field of ...
3
votes
0answers
69 views

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 ...
4
votes
0answers
156 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{...
5
votes
1answer
163 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 ...
9
votes
1answer
288 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 ...
2
votes
0answers
149 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 $G$ ...
1
vote
0answers
36 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 ...
2
votes
1answer
106 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 $m \cap k[X_i] \ne 0 , \forall i \in I$ ...
1
vote
1answer
100 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 $...
2
votes
1answer
75 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....
4
votes
1answer
102 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, ...
3
votes
3answers
320 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 $\...
0
votes
1answer
162 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 ...
4
votes
0answers
168 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, ...
9
votes
1answer
256 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 $D$ ...
0
votes
1answer
140 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 $...
0
votes
1answer
61 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\...
2
votes
1answer
75 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 ...
3
votes
1answer
139 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 \\ \...
2
votes
0answers
139 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, ...
3
votes
1answer
167 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 ...
6
votes
1answer
194 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 ...
0
votes
0answers
57 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 $X$. ...
1
vote
1answer
123 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{...
2
votes
0answers
385 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 ...
28
votes
1answer
544 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 ...
11
votes
3answers
588 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 ...
4
votes
0answers
285 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 ...
7
votes
0answers
159 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 ...
4
votes
1answer
333 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 ...
4
votes
0answers
183 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 ...
4
votes
1answer
312 views

What kinds of limits does localization of commutative rings reflect?

Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...
1
vote
0answers
155 views

Local cohomology commuting with fiber

Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$). Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...
11
votes
2answers
312 views

Reflective Localizations vs. categories of local objects

Given a category $\mathcal{C}$ and a set (let's not bother with size issues here) $\mathcal{W} \subseteq \text{Mor}(\mathcal{C})$ we may form the category $\mathcal{C}[\mathcal{W}^{-1}]$ obtained by ...
6
votes
1answer
162 views

Monoidal structure on simplicial sheaves

Let $\mathcal{C}$ be a site and let $sPh(\mathcal{C})_{proj}$ be the category of simplicial presheaves equipped with the projective model structure. This category is a closed monoidal model category (...
9
votes
1answer
390 views

Example of a saturated class of morphisms which is not _obviously_ saturated?

By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists ...
6
votes
2answers
303 views

Localizations of model categories and $\infty$-categories

I am interested in the relation between Bousfield localizations of model categories and localizations of $(\infty,1)$-categories. According to Hirschhorn's book we can form the left Bousfield ...
37
votes
2answers
1k views

What is an infinite prime in algebraic topology?

The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...
2
votes
1answer
110 views

Near-ring localizations

Are there any known results on localization for near-rings (i.e., "rings" with non-abelian addition and only one-sided distributive law)? The books on near-rings I checked don't mention this topic at ...
3
votes
2answers
436 views

Localization of symmetric monoidal category

Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T \...
21
votes
9answers
3k views

Why localize spaces with respect to homology?

A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...
10
votes
1answer
245 views

Localizing 2-categories about a single morphism

This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and ...
3
votes
3answers
548 views

Dimension of a ring after localization

Let $R$ be a Noetherian domain of dimension $\ge 1$. Let $\mathfrak{p}_i$, $i = 1, 2, ...$ be prime ideals of height one. Let $T = R[[X]]$ with $X$ is a indeterminate. For each $i \ge 1$ we set $\...
1
vote
0answers
928 views

Sufficient condition for local martingale property of stochastic integral

Is the following correct and/or a (simple) known result? Let $X$ be a local martingale and $H$ an integrand for $X$, such that the stochastic integral $\int H\cdot dX\ge x$ for some random variable. ...
2
votes
1answer
272 views

Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$

Let $R=\mathbb{Z}_p[[X]]$ where $\mathbb{Z}_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R_{(p)}$ $(R$ localised at the ideal $pR)$ $?$
1
vote
0answers
80 views

phase prediction of wavelet coefficients for 1D signal [closed]

I was reading a paper 'A Flexible Framework for Local Phase Coherence Computation' (article URL) on predicting phases of wavelet coefficients across 3 consecutive scales in the 1D case, and I'm trying ...