12
votes
1answer
209 views

localizing subcategories of $HF_p$-local spectra

This entire question takes place in the $HF_p$-local category of $p$-local spectra, i.e. the essential image of $HF_p$-localization on the stable homotopy category. $HF_p$ itself is in there, and of ...
3
votes
0answers
202 views

Fields in Stable Homotopy Theory

It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these ...
2
votes
1answer
237 views

Counterexamples to Smallness of Harmonic Spectra

It is a theorem of Neil Strickland's that the category of harmonic spectra (i.e. the category of $p$-localized spectra localized at the infinite wedge of Morava K-theories) has no small objects. That ...
8
votes
1answer
307 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 ...
4
votes
0answers
109 views

Bousfield Lattices for which Minimal Objects Coproduct to Sphere Object

Is it known what conditions we require of a stable homotopy category to have $\langle S\rangle = \coprod\limits_{\mathbb{N}}\langle K(n)\rangle$, where $\langle K(n)\rangle$ is some minimal Bousfield ...
5
votes
1answer
220 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 ...
6
votes
2answers
354 views

Chromatic convergence of E(n)-localized homotopy categories

Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy ...
1
vote
2answers
219 views

Relationship of Bousfield Classes of Morava K-theories

Suppose we have $\langle K(n)\rangle$ and $\langle K(n-1) \rangle$ for some fixed prime $p$. do we know whether or not $\langle K(n) \rangle \geq \langle K(n-1) \rangle$ or $\langle K(n-1) \rangle ...
2
votes
3answers
270 views

Bousfield Classes

This question has a few parts: 1) Is the Bousfield class of $\langle E\rangle$ the class of $E$-acyclics, i.e. $\langle E\rangle=\left\{ X:E\wedge X=0\right\}$ or is it the class of spectra which are ...
5
votes
2answers
304 views

Does a finite suspension spectrum make a space finite?

Suppose that $X$ is a space whose suspension spectrum $\Sigma_+^\infty(X)$ is dualizable in the stable homotopy category. I believe this is equivalent to saying that $\Sigma_+^\infty(X)$ is (weakly) ...