# Questions tagged [bousfield-localization]

The bousfield-localization tag has no usage guidance.

The bousfield-localization tag has no usage guidance.

33
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Let us work concretely at the prime 3. How does $TMF \wedge X \simeq E(2)$ for X a finite type 0 spectrum imply that $TMF$ and $E(2)$ have the same Bousfield class?

6
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1
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201
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We fix $p$ prime and $n$ a natural number. We let $K(n)$ be the $2(p^{n}-1)$-periodic Morava $K$-theory, i.e. $K(n)_*=\mathbb{F}_p[v_n^{\pm 1}]$ with $|v_n|=2(p^n-1)$. I distinctly recall that we ...

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I'm in the process of writing an expository paper on complex K-theory and Snaith's theorem; the proof of Snaith's theorem that I'm following along (located at http://math.uchicago.edu/~amathew/snaith....

9
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282
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Let $S$ be a finite $p$-group and $K$ a compact Lie group, in the paper A Segal conjecture for $p$-completed classifying spaces, it is said that the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\...

3
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162
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Let $\mathbb{Z}/p^{\infty}$ denote the Prufer group. By $p$-completion properties, it follows that $(B\mathbb Z/p^{\infty})^{\wedge}_p\simeq K(\mathbb{Z}^{\wedge}_p,2)\simeq(BS^1)^{\wedge}_p$. But, ...

7
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$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ Suppose $i:A\to B$ is a map of $E_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod_B\to\Mod_A$ by ...

6
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211
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When studying the stable homotopy of $BG^{\wedge}_p$, with $G$ a finite group, authors know that this abuse of notation is not dangerous because $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^...

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246
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Bousfield p-completion on spaces is a functor $(-)^{\wedge p}$ whose main property is that a map $f:X\rightarrow Y$ induces an isomorphism $f_{\ast}:H_\ast(X,\mathbb{F}_{p})\rightarrow H_\ast(Y,\...

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119
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Let $\mathbb{S}$ be the sphere spectrum. We can localize the category of based spaces, $Top_*$ at a homology theory, and hence at $\mathbb{S}$.
Equipping $Top_*$ with the Quillen model structure (...

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75
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Let $\mathcal{M}$ be a model category and $S$ a set of cofibrations between cofibrant objects. Then every $S$-local object has the right lifting property with respect to $S$. The converse does not ...

3
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206
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The topic of Bousfield localizations has a lot of literature which has on most of the occasion some tameness assumption on the presentability of the model category. Recently I have been trying to ...

2
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232
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By lemma 4.9 in Dugger-Hollander-Isaksen, a hypercover is defined as an augmented simplicial object $U_\bullet\to X$ in the category of simplicial presheaves such that each $U_n$ is a coproduct of ...

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Let $\mathcal{M}$ be a pointed model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, ...

5
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1
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211
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Let $L_{Nis}(sPre(Sm_S))$ be the Nisnevich localization of the category of simplicial presheaves,
how to see that whether $\mathbb{A}^1$-projections $\mathbb{A}^1\times_S X\to X$ are closed under ...

5
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109
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By Ohkawa's theorem, the Bousfield lattice $B$ (of the $\infty$-category of spectra) is a small, complete lattice with $2^{\aleph_0} \leq |B| \leq 2^{2^{\aleph_0}}$ (the exact cardinality is an open ...

7
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281
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It seems to me that although homological localization (i.e. formally inverting $E$-homology equivalences for some $E$) is a reasonable thing to do to a spectrum, it's a pretty brutal thing to do to a ...

15
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840
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Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ?
It is well known to exists ...

3
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250
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Let $E$ be a generalized cohomology theory. Let's agree that $E$ satisfies property $(*)$ if for any two finite CW complexes with isomorphic homotopy groups their $E$-localizations also have ...

6
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I know very little about algebraic topology, and more about $k$-linear stable $\infty$-categories (i.e. homological algebra).
Given an abelian group $A$, there is the Eilenberg-Mac Lane spectrum $HA$,...

15
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1
answer

580
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If $X$ is a spectrum with trivial (integer-valued) homology groups, does it have to be weakly-equivalent to a point?
This is easy to prove for connective spectrum, as a Hurewitz-type argument is then ...

5
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2
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355
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Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}\to X_n), (q_{n+1}: Y_{n+1}...

6
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199
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Since my question is very specific let me introduce the context. I am trying to apply derivator theory to stable homotopy theory.
Theorem 3 of the following preprint by Franke
https://pdfs....

6
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178
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Let $M$ be a finite spectrum, so that $L_{K(n)} M = M \wedge L_{K(n)} S$. Recall that (a weak version of) the chromatic splitting conjecture states that the chromatic attaching map $L_{n-1} M \to L_{n-...

18
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2
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The Brown-Comenetz dualizing spectrum $I_{\mathbf{Q/Z}}$ is not detected by very many spectra: it is $BP, \mathbf{Z}, \mathbf{F}_2, X(n)$ for $n\geq 2$, and even $I_{\mathbf{Q/Z}}$-acyclic. However, ...

8
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408
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I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those ...

5
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253
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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 ...

5
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235
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Let $\mathcal T$ be a triangulated category and $\mathcal C$ a thick triangulated subcategory. We consider the Verdier quotient $\mathcal T/\mathcal C$.
Is there a bijective correspondence between ...

6
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1
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149
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Suppose $E$ is a spectrum and $p$ is a prime. We can then $(H\mathbb{Z}/p)_*$-localize to obtain $L_{H\mathbb{Z}/p}E$. Is it true that the natural map $L_{H\mathbb{Z}/p}E\rightarrow\text{holim}_n(L_{H\...

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428
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My first question in this forum.
Suppose we have a Quillen equivalence $L: M\leftrightarrow N: R$ between two model categories (say left proper cofibrantly generated cellular or locally presentable). ...

3
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89
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I have been looking around for examples of $R$-bad spaced in the sense of Bousfield and Kan. In their book "Homotopy limits, completions and localizations] they give several examples of such spaces ...

6
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477
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The notion of a (left, say) Bousfield localization of a model category doesn't seem to be invariant under Quillen equivalence. There are a lot of things that could go wrong. But I don't know any ...

2
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1
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230
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I'm learning about Bousfield localizations. For a triangulated category satisfying some axioms, a Bousfield localizations can be described as an idempotent functor $L:D \to D$.
I thought there is a ...

12
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2
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666
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I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them.
First definition: Let $\mathbf{...