The (very nice) Bousfield Lattice of Harmonic Spectra

So this is my first post here on Chromotopy (thanks to Matt Pancia for making me aware of this blog, and Eric Peterson for letting exercise my typing fingers!). Just thought I might mention a little doodle about Bousfield lattices that occurred to me recently. It’s nothing too complicated, and it’s kind of cute, so I’ll include the proof. This is something that I realized while I was futzing around with general Bousfield lattices, notions of Stone duality (i.e. categorical anti-equivalences between certain species of lattice and certain species of topological space). Ultimately, I suspect it follows from one or two of the many beautiful theorems hidden away in papers like “Axiomatic Stable Homotopy Theory” and other stuff by Hovey, Strickland and Palmieri.

Here’s the point: in Mark Hovey and Neil Strickland’s beautiful (and incredibly dense) paper “Morava K-theories and Localisation” they show that the Bousfield lattice of the $E(n)$-local category of spectra is precisely the finite Boolean algebra generated by $n$ objects, specifically the Bousfield classes of Morava K-theories, $\langle K(n)\rangle$.  The natural next question, for me at least, is what the Bousfield lattice of the harmonic category (i.e. the $p$-local stable homotopy category localized at the infinite wedge of Morava K-theories $\bigvee_n K(n)$) looks like (let’s denote this category by $\mathcal{H}$).  And it turns out, it’s precisely what one would expect! That is, it’s the colimit of the Bousfield classes of the $E(n)$-local categories, i.e. the infinite Boolean algebra generated by a countable number of minimal elements.

How do we know this? Well, what we really want to do is determine the harmonic Bousfield class of a harmonic spectrum $X$, which I’ll denote by $\langle X\rangle_H$.  What does that really mean? It means that $\langle X\rangle_H=\{ Y\in\mathcal{H}:Y\wedge X\simeq \ast\}$.  If you’re quick (which I am not), you certainly already see how to do this. If you’re like me, it might take you a few days, so since it’s not that big of a deal, you should just keep reading.

Let $supp(X)=\{K(n):X\wedge K(n) \not\simeq \ast\}$ and $cosupp(X)=\{K(n):X\wedge K(n)\simeq \ast\}$ .  What we’re going to do is show that $\langle X\rangle_H=\bigvee_{K(n)\in supp(X)} \langle K(n)\rangle$.  Suppose that $Y\in\langle X\rangle_H$.  So, $Y\wedge X\simeq\ast$, and imortantly, $Y\wedge X\wedge K(n)\simeq\ast$.  In other words, $Y\in\langle X\wedge K(n)\rangle_H=\langle K(n)\rangle_H$. Where this last equality follows from the fact that $K(n)\wedge X=\vee\Sigma^{d_i} K(n)$ for some collection of $d_i$. Hence, $Y\in\langle \bigvee_{supp(X)}K(n)\rangle=\bigvee_{supp(X)}\langle K(n)\rangle$.

Now, suppose that $Y\in\langle \bigvee_{supp(X)}K(n)\rangle$.  Then we want to show that $Y\wedge X\simeq\ast$.  So, for every $K(n)\in supp(X)$, it’s clear that $Y\wedge X\wedge K(n)\simeq\ast$.  But for every $K(m)\in cosupp(X)$, we have that $Y\wedge X\wedge K(m)\simeq\ast$ also! So, since $X\wedge Y$ is harmonic, but $X\wedge Y\wedge K(n)\simeq\ast$ for every $n$, it’s also contractible!

One remark: I’ve neatly brushed under the rug here that colocalizing subcategories (e.g. harmonic spectra) may not be closed under smash product (I don’t think).  So, I really should have written $\wedge_H$ or something every time I wrote $\wedge$, but it works out fine if you just assume that after I took the smash product, I localized at the harmonic category again.

Anyway, the reason I think this is interesting is that this lattice has a really nice Zariski spectrum, and I first started looking at it hoping that it might approximate the Bousfield lattice of $p$-local spectra in some nice way.  But as far as I can tell, it’s just too simple (you never get something for nothing, I suppose).  I was hoping to make something of some ideas of Jack Morava regarding sheaves of spectra over the Bousfield lattice, but it may be that Bousfield lattices are just too coarse to do this kind of thing, or rather, tensor-triangulated categories are too coarse. We really need to rewrite the HPS paper I referenced above in the language of infinity categories and derived algebraic geometry.

It’s also interesting to note that the little proof above implies that the telescope conjecture hnews inside the category of harmonic spectra.  Obviously this doesn’t really say anything about the telescope conjecture outside this category. It’s also known, for instance, that the telescope conjecture hnews inside the category of $BP$-local spectra. However, the Bousfield lattice in that category is not known. It’s structure is the content of a conjecture from Doug Ravenel’s well known ‘84 paper that is still open!

I hope to write more later about Stone duality for lattices/locales and Bousfield lattices in general.

:)