# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### Free algebra and ring of quotients [closed]

I am reading of example T.Y.Lam 'A first Course in Noncommutative Rings': Let $R=\mathbb{Z}\langle x,y\rangle/(y^2,yx)$. To work with $R$, we shall confuse $x,y$ with their images in $R$. Thus,we ...
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### 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 ...
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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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### 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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### 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 properties:...
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### 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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### 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: http://imrn.oxfordjournals.org/content/2002/7/351.full....
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### 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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### 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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### 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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### (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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 \mathrm{Hom}_{\...
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 ...
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 \$[X,Y]_\ast=...