The localization tag has no usage guidance.

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### 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 ...

**3**

votes

**1**answer

224 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 ...

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126 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 ...

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votes

**2**answers

195 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 ...

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votes

**1**answer

89 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 ...

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votes

**1**answer

307 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 ...

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200 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 ...

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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 ...

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**1**answer

95 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 ...

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votes

**1**answer

221 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 ...

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### 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 ...

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**1**answer

211 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 ...

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**3**answers

350 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 ...

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**0**answers

384 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. ...

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votes

**1**answer

256 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 $pR)$ $?$

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68 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 ...

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**0**answers

180 views

### Weak equivalences of left Bousfield localizations

Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1.
If necessary, the model structures can be assumed to be simplicial, ...

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**1**answer

204 views

### p-complete Z_p-modules

Let $D(\mathbf{Z})$ be the derived category of abelian groups, and let $D(\mathbf{Z}_p)$ be the derived category of modules over the p-adic integers. Bousfield localization gives a full subcategory of ...

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**2**answers

254 views

### Cool Examples of Localisation in Triangulated Cats Besides the Usual

In the theory of triangulated categories there is a hefty literature on localisation -- the most common example in algebra being (variants of) localising the homotopy category of chain complexes over ...

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**1**answer

194 views

### is every finitely n-presented (S^{-1})R-module a localization of a finitely n-presented R-module?

Let S be a multiplicative set in a ring R. We can see that every finitely generated $(S^{-1})R$-module is a localization of a finitely generated R-module.
Then, more generally, is every finitely ...

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**1**answer

202 views

### Localisation of inclusion functors

Let $\mathcal C$ be a category and suppose $\cal B \subseteq C$ is a full subcategory. Let $i \colon \mathcal B \longrightarrow \cal C$ denote the inclusion functor. Suppose that $S \subseteq ...

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244 views

### Localisation of injectives

When working with injective modules, one bad thing is that they do not necessarily behave well with respect to localisation. Consider a commutative ring $R$ and have a look at the following ...

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**1**answer

145 views

### Does Wolbert's derived equivalence between $E_*^R$-local $R$-modules and $R_E$-modules come from a Quillen equivalence?

Let $R$ be a ring spectrum (in the world of EKMM $S$-modules) and let $E$ be a smashing $R$-module. Denote by $R_E$ the $E_*$-localization of $R$. By a theorem of Wolbert (Theorem 2 in Classifying ...

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334 views

### Is the localisation of a product of categories the product of the localisation?

Let $\cal C, \cal D$ be model categories. Hovey says in his monograph "Model Categories" that the homotopy category $\operatorname{Ho}(\cal C \times D)$ is isomorphic to $\operatorname{Ho}(\cal C) ...

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**1**answer

228 views

### Bousfield localization before and after taking homotopy

The ncatlab says:
Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left ...

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**2**answers

249 views

### Localisation in a quasi-category

Let $W$ be a family of arrows in a category $\mathcal{C}$, there is a nature notion of localisation w.r.t. $W$. And if $W$ satisfies some nice properties, we have calculus of fraction.
Now consider ...

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298 views

### Intersection of localization with finitely generated subalgebra of fraction field

Let $R$ be a (commutative) noetherian integral domain. Let $I$ be a prime ideal of $R$. Let $S$ be a finitely generated $R$-subalgebra of $\mathrm{Frac}(R)$.
Is $S \cap R_I$ necessarily finitely ...

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578 views

### How to prove Arnold Conjecture without using S^1 localization?

By the Arnold Conjecture, I mean the following statement:
Let $M$ be a closed symplectic manifold, and $\phi:M\to M$ a Hamiltonian symplectomorphism with nondegenerate fixed points. Then $ \# ...

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votes

**1**answer

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### Voevodsky's proof in any characteristic (for motivic and Chow)

Regarding the published version of "Motivic cohomology groups are isomorphic to higher Chow groups in any characterstic" (IMRN) available at:
...

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**1**answer

389 views

### Smashing localizations in the category of spectra

Let $E$ be a spectrum. Then $E$ determines an idempotent localization functor $L_E: \mathrm{Sp} \to \mathrm{Sp}$ sending each spectrum to its $E$-localization.
The functor $L_E$ generally does not ...

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### Is there a notion of a “model category which admits left Bousfield localization?”

At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which ...

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**1**answer

308 views

### Is being principal a local property?

Let $R$ be a number ring and a Dedekind domain. We have the following result:
For every ideal $I\subset R$ $$ I = \bigcap_P I_P $$ where $I_P$ denotes the localization of $I$ at $P$ and the ...

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### The multiplicative system in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category. In the 1973 paper "Note on monoidal localisation" by Brian Day, the multiplicative system of morphism in $\mathcal{C}$ has been discussed. See also ...

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173 views

### Eigenvector localizaiton

I have raised this sort of question before but I think that now I've found a better term for the subject, one which might ring more bells for people - hence the repost. Hope you won't be too angry ...

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### Is primary decomposition still important?

On p.50 of Atiyah and Macdonald's Introduction to Commutative Algebra, in the introduction to the chapter on primary decomposition, it says
In the modern treatment, with its
emphasis on ...

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**1**answer

448 views

### (Co)localization of the derived category

Let me start saying that a similar question can be stated for general locally Noetherian Grothendieck categories but I state it for categories of modules as it is simpler. So we fix a right Noetherian ...

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340 views

### Localization of quotient rings of polynomials

Working on some Bezout's theorem examples I arrived at a point where I need
$$(\mathbb{K}[x]/(x^2))_{(x)} = \mathbb{K}[x]/(x^2)$$
(i.e. localize don't do anything)($\mathbb{K}$ alg. closed and nice ...

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votes

**1**answer

259 views

### Is there an obvious reason why p-localization of spectra is a finite localization?

Is there an obvious reason why $p$-localization of spectra is a "finite" localization in the sense of Haynes Miller? In other words, is there an obvious reason why the localizing subcategory (of the ...

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votes

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1k views

### Total ring of fractions vs. Localization

Let $R$ be a commutative ring and denote by $K(R)$ its total ring of fractions, the localization of $R$ with respect to $R_{\mathrm{reg}}$. For every multiplicative subset $U \subseteq R$ there is a ...

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vote

**1**answer

220 views

### Are these connecting homomorphisms commutative?

Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative?
In other words, is the following statement true?
If it is true, then, how can one prove it?
...

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votes

**5**answers

854 views

### Noncommutative Localization of a Ring : Complete Construction

I've been looking for the following construction in the literature, but I've only been able to find (very) partial proofs or proofs of special cases.
Let $R$ be a non-commutative ring and $S$ a ...

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292 views

### Is it possible to define the notion of a localization of a category without reference to a set of morphisms, $S$?

Let $\mathcal{C}$ Be A Category and $S$ a class of morphisms (let us call these weak equivalences) of $\mathcal{C}$. One often defines the localization of $\mathcal{C}$ with respect to $S$ is the ...

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668 views

### Localization of a symmetric monoidal category at a single morphism

Let $C$ be a symmetric monoidal category, and $f : x \to y$ be a morphism in $C$. I would like to construct the localization $C_f$ explicitly, which solves the universal property
...

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### Ring properties which can be checked on sufficiently many Ore localizations

Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of ...

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557 views

### Absence of Maps Between p-local and q-local spectra

Suppose $X$ and $Y$ are spectra (or homotopy classes thereof) such that $X$ is p-local and $Y$ is q-local, for primes $p\neq q$. Is it indeed true then, and if so how would one show that ...

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### Elements in a localization - category theoretic approach

This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa = 0$ only using the ...

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### Computing morphisms in localizations of $K(B)$

Let $B$ be an additive category (a small one; one can assume that it is a $\mathbb{Q}$-category, yet not much else is known about it). Given a set of objects $S$ of $K^b(B)$ (or $K(B)$), I consider ...

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### Topological Localization of (the simply-connected cover of) SO or Spin

This is essentially a (cw) reference request, because it seems like the sort of thing that should have been looked at already.
Setting aside, for now, how to think what the localization of a general ...

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**1**answer

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### Projective modules over semi-local rings

Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. If the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$ then $M$ is free.

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### Cross correlation detection in binary Hamming distance

Given two long binary strings of length N, it's easy to find the Hamming distance between them. If you're allowed to cyclically shift one of the strings, you'll get N different Hamming distances when ...