Questions tagged [localization]
The localization tag has no usage guidance.
171 questions
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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 ...
- 63.1k
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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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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 ...
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Rig of fractions, including zero denominators
For some integral domain $R$, one forms the field of fractions $R^*$ by considering (equivalence classes of) formal pairs {$r/s : r \in R, s\in R\backslash 0$} and defining $+$ and $*$ as you'd expect ...
- 1,872
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Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category?
Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the ...
- 19.6k
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Controlling Reflective Subcategories and Localizations
Localizations are an extremely important part of modern homotopy theory. Both the category of spaces an spectra have a plethora of interesting localizations: at a fixed prime, rational, with respect ...
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Localizing an arbitrary additive category
Under which conditions localizing an additive category by some class S of morphisms yields and additive category? It seems easy to define certain addition on morphisms if we fix their representatives ...
- 16.9k
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Checking locally whether a homomorphism is a localization
All rings below are commutative with $1$.
Suppose $A\subset B$ is a subring and that $A\rightarrow A'$ is a faithfully flat ring homomorphism. [You may assume the rings are actually ${\mathbb C}$-...
- 2,083
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Flatness of module
$A\rightarrow B$ a ring homomorphism, $N$ a $B$-module which is flat over $A$. $\mathfrak{q}\subset B$ a prime ideal, $\mathfrak{p}\subset A$ its contraction in $A$. Then is it true that $N_{\mathfrak{...
- 2,857
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Localization of module
M an A-module, $S\subset A$ a multiplicative subset. Is it possible for $S^{-1}M$ to have an $S^{-1}A$-module structure satisfying $\frac{a}{1}\cdot\frac{m}{1}=\frac{am}{1}$ other than the "usuall" ...
- 2,857
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Reasons to believe Vopenka's principle/huge cardinals are consistent
There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...
- 19.6k
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Localization of power series and module structure
Let $R=\mathbb{Q}[X,Y]$ be the polynomial ring of two commuting variable.
Let $S$ be the multiplicative subset of $R$ generated by homogeneous linear polynomials.
Let also $\widehat{R}$ be the ring of ...
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Spectra and localizations of the category of topological spaces
Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces
using some kind of localization combined with other categorical ...
- 37.8k
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Atiyah Bott localisation applied to Euler characteristic
Suppose we have a torus action on a compact oriented manifold M. Assume the action has isolated fixed points. Why is it that the equivariant Euler class of the normal bundle at the fixed point (i.e. ...
- 3,383
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Localizing at the primitive polynomials?
For any UFD $R$, the concept of a primitive polynomial (gcd of the coefficients is 1) makes sense in $R[x]$. The product of two primitive polynomials is primitive (Gauss's Lemma), and certainly 1 is a ...
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is localization of category of categories equivalent to |Cat|
It might be a stupid question.
Suppose There is a category of categories,denoted by CAT,where objects are categories, morpshims are functors between categories
Take multiplicative system S={category ...
- 5,515
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Equality of elements in localization via universal property
I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. A very nice example for this ...
- 63.1k
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What is the localization of Q[x]/(x) at 0
Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??
I think the result is
Q[x]/...
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Localization(s) of Categories
I've been trying to read a paper by Krause and came across a strange (to me, of course) notion of localization. After looking around for a long time, and then finding this on his site, I see that ...
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