User jon beardsley - MathOverflow

Non-commutative Formal Group Laws

2013-04-29T12:58:52Z

Does anyone know of a good, complete reference for non-commutative formal group laws (i.e. construction of a "Lazard ring," discussion of non-commutative formal groups, perhaps some discussion of their moduli stack)? I seem to recall being told that there were at least a few South American mathematicians doing some really nice work on this sort of stuff, but don't know their names and haven't been able to find it. Thanks for any references on this! :)

Localization at Infinite Wedges of K-theories or BP

2012-08-20T22:41:42Z

This is basically a reference request. Does anyone know if the structure of the homotopy category of spectra (or maybe just the model, i.e. w/o the homotopy, category), localized at infinite wedges of Morava K-theories, or BP, is either well described somewhere or somehow stupid and uninteresting? This question is motivated by i) I believe Morava K-theories and the telescope spectra T(n) are the same BP-locally, i.e. a sort of telescope conjecture, and ii) wondering if localizing at BP or maybe the wedge of all the Morava K-theories would somehow pick out all the chromatic information in the category of spectra. That is (and I think my details will be off because I haven't thought about this in a little bit), there is some idea in the derived category of a Noetherian ring that we have these localization functors which correspond to prime ideals of the ring, and that maybe localizing at BP is somehow localizing the stable homotopy category at the $I_n$ ideals or something. Any commentary is appreciated.

Thanks!

Answer by Jon Beardsley for Localization at Infinite Wedges of K-theories or BP

2013-04-05T22:21:46Z

I mentioned that I proved this in the comment above but am "answering" just for closure. A link to the proof is here:

http://chromotopy.org/?p=1110

Mayer-Vietoris Sequence for Arbitrary Bicartesian Square of Spectra

2013-03-01T00:40:45Z

Can anyone tell me if there is a Mayer-Vietoris sequence for an arbitrary homotopy pushout (hence homotopy pullback) of spectra and an arbitrary (co)homology theory. If this comes from some easy way of writing down a pushout/pullback as a fiber sequence, it'd be really cool to see that spelled out (as if I were a baby). Also, it would also be really neat to know if there are general conditions on a model category or on the (co)homology theory to make this true.

Thanks in advance. :-)

Filtration on Smash Product of Cofibers

2013-02-19T02:26:38Z

I have seen some similar questions to this one on here recently, so I hope this isn't redundant. Basically, suppose I have two cofiber sequences of spectra (or perhaps just work in some general homotopy category or something) $X\overset{f}\to Y\to Cf$ and $X'\overset{g}\to Y'\to Cg$. I'd like to look at an induced filtration on $Cf\wedge Cg$ along the lines of the bottom level being $X\wedge X'$, the middle level being something like $(X\wedge Y')\cup (Y\wedge X')$ and the top level being $Y\wedge Y'$, or something along those lines. Or I guess I should say I'd like to have filtration quotients that look like that. Honestly, I'd like to do this all in a cellular context. Intuitively this all seems pretty obvious, but does anyone have a good reference where such a filtration is discussed really rigorously, or a good way to think about it?

Thanks, as always.

Answer by Jon Beardsley for Filtration on Smash Product of Cofibers

2013-02-19T17:39:51Z

So it seems to me that, using Peter May's paper above, for two distinguished triangles $X\to Y\to Z$ and $X'\to Y'\to Z'$, we have a 'filtration':

$$Z\wedge Z'\overset{f_1}\leftarrow Y\wedge Y'\overset{f_2}\leftarrow X\wedge X'$$

where the cofiber of $f_1$ is $Z\wedge Z'\to\Sigma V=\Sigma(X\wedge Y'\cup_{X'\wedge X}X'\wedge Y)$ and the cofiber of $f_2$ is just what it is, though Peter May's paper denotes it by $W$ and gives other distinguished triangles as well as several Cartesian squares that $W$ fits inside of. That way, we can attempt to understand what it is. Perhaps there is a better way to combine these things, but this is the best way I've determined thus far.

Connection between complex orientations and R-orientations for a ring spectrum R?

2013-02-12T19:21:09Z

We have a well defined notion of complex orientation for a spectrum (coh. theory) $E$, that is, we have a class $x_E\in \tilde{E}^2(\mathbb{C}P^\infty)$ which restricts to identity along the inclusion $\mathbb{C}P^1\hookrightarrow \mathbb{C}P^\infty$. Is there a connection between this notion of "complex orientation" and the notion of a Thom spectrum being $R$ oriented with respect to a ring spectrum $R$ in the sense of ABGHR? That is, for a Thom spectrum $Mf$ associated to a map $f:X\to BGL_1\mathbb{S}$, and a ring spectrum $R$, we say that $Mf$ is $R$-oriented if the composition $X\to BGL_1\mathbb{S}\to BGL_1R$ is null. If we consider $MU$ to be the Thom spectrum associated to some map (I'm not sure which it should be, but I suspect this is the way it's done) $BU\to BGL_1\mathbb{S}$, can we rephrase the notion of complex orientation in this language?

Thanks!

Calculations of Homotopy Coinvariants

2013-02-01T19:45:58Z

Does anyone know of a reference in which homotopy coinvariants of some coaction in the category of spectra are explicitly calculated? Particularly in the case of Hopf-Galois extensions of ring spectra.

Thanks!

-----(EDIT)------ I should mention that I later realized the above concept is not in John Rognes' monograph. It is in Kathryn Hess' paper on homotopical Hopf-Galois extensions and Fridolin Roth's recent dissertation. My apologies to anyone who went looking in Rognes' paper.

Forcing in Homotopy Type Theory

2013-01-14T05:11:17Z

I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? Presumably we do homotopy type theory in some $(\infty,1)$-topos, so we can axiomatize the notions accordingly? Does anyone know of a reference for this kind of thing if it does exist or makes sense?

Thanks

Counterexamples to Smallness of Harmonic Spectra

2013-01-08T00:06:52Z

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 is to say that there is not a single spectrum $X$ in the category of harmonic spectra such that $[X,\bigvee X_i]=\bigoplus[X,X_i]$. Take a finite spectrum in this category (for instance the sphere spectrum $\mathbb{S}$). Does anyone have an example of a coproduct in the category of harmonic spectra for which the above is NOT true? While Strickland's proof makes sense, it seems completely amazing to me. What HAPPENS to the sphere and the cells of finite spectra when we localize? It seems that if something were small to begin with, i.e. the above homotopy groups factored globally, killing some spectra shouldn't change this! Note that the same question can be asked of the $BP$-local category.

Are there conditions (other than finiteness) that we can put on the coproduct to make the statement true when the object on the left is finite (i.e. in the thick subcategory generated by the sphere)?

Connection of X(n) spectra to formal group laws

2012-12-17T23:26:10Z

In the proof of the Nilpotence Theorem, or at least in Ravenel's account of it in his Orange Book, a sequence of spectra are used, denoted $X(n)$ with $X(0)=\mathbb{S}$ and and $X(\infty)=MU$ such that $\langle X(n)\rangle\geq\langle X(n+1)\rangle$. These are the Thom spectra associated to the map $\Omega SU(n)\to BU$. They are homotopy equivalent to $MU$ up through degree 2n-1. They all have associated Hurewicz maps $h(n):\pi_\ast(R)\to X(n)_\ast(R)$ for $R$ a finite ring spectra. We are interested, for the Nilpotence Theorem, in determining when $h(n)(\alpha)=0$, ultimately for $n=0$. To this end, Ravenel proves that if $h(n+1)(\alpha)=0$ then $h(n)(\alpha)$ is nilpotent.

So, from this theorem we know then that any of the $X(n)$ spectra detect nilpotence just as well as $MU$. The nice thing of course about $MU$ (or one of the many nice things) is that it has at least one other interpretation (i.e. aside from its geometric interpretation as a Thom spectrum). This is of course that $MU_\ast$ determines formal group laws over rings, and you have all of this amazing stuff happen. Do you have any similar such interpretations for these $X(n)$ spectra, or are they ONLY geometric in nature? Or I guess, to rephrase, does anyone KNOW of any other ways of thinking about these things?

Thanks!

Compact MU or BP Modules

2012-12-14T21:41:32Z

Is there a classification of the compact MU or BP modules in any category of spectra? Can the periodicity theorem be finagled to give a MU-module structure on finite spectra?

Is there a category of spectra whose Bousfield classes are Bousfield idempotent?

2012-12-04T22:27:03Z

It is common when looking at the Bousfield lattice of a tensor-triangulated category to consider a sublattice that one might denote by $DL$. This is the sublattice of Bousfield classes $\langle X\rangle$ such that $\langle X\rangle=\langle X\wedge X\rangle$ (it's called $DL$ because it turns out to be distributive). Let's suppose we only care about the homotopy category of spectra. Is there anything which prohibits us then from considering the full subcategory of objects whose Bousfield classes are idempotent in the above way? Or rather, a more sensible question is, has anyone studied whether or not this category has any of the nice properties that the homotopy category of spectra does? Does it have any nice closure properties? Obviously it is closed under smash and wedge.

Generators of Thick Subcategories

2012-11-30T23:29:14Z

Suppose we are given a thick subcategory of the compact objects in the homotopy category of modules over a ring spectrum $R$. Are there conditions we can place on $R$, or on the category (compact) $R$-modules to ensure that every thick subcategory has a single generator? It seems that, given the existence of finite Bousfield localizations, this would be solved by showing that the associated finite colocalization, which is a functor to the given thick subcategory, preserves compactness. But it doesn't seem that in general this should be true. Certainly localization functors do no always preserve compactness.

My best idea at the moment is to show that finite localizations of compact objects are compact, hence the fiber of the localization morphism $X\to\mathcal{L}^{fin}X$ is a fiber of compact objects, hence compact itself. Then one could apply this to $R$ itself. I am not sure if all of those statements are valid however, and am worried that there are some counterexamples I am not aware of.

Right Notion of Localizing Subcategory in Quasicategory

2012-11-14T23:04:56Z

Given a stable quasicategory $C$, what are the conditions required of a subcategory for it to descend to a localizing subcategory in $Ho(C)$? Is it enough for it to be reflective? Perhaps exact and reflective?

Thanks, Jon

Periodicity Theorem for D(R)

2012-11-01T20:15:20Z

For a derived category of a Noetherian ring (or perhaps more generally), can we talk about a Periodicity Theorem? We have Thick Subcategory Theorems and Nilpotence Theorems (HPS 91) for D(R), and in some more general cases, but I haven't seen, in my limited review of the literature, a theorem describing how such data might "stratify" the endomorphisms of the unit object (in this case I guess R). Or, in a similar vein, finding specific maps whose cofiber lies in perhaps the "next higher" level. Perhaps this is connected to the fact that for a ring, unlike in the case of finite p-local spectra, there isn't really a linear ordering of primes, necessarily. Anyone know of any work on this idea? Or perhaps know a straightforward answer?

Thanks!

Monoidal Model Categories with Suspension Functor

2012-10-29T19:47:18Z

This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model category of spaces, i.e. obtained by inverting the suspension functor, or something along those lines, what is this process called? What is the data on a model category that I need to make this happen? Is this structure available in $(\infty,n)$-categories as well? I also think about this in terms of going from R-algebras to R-modules (perhaps in some derived sense), i.e. taking a sort of tangent space or tangent category. Is there a standard framework for such things? In the case of model categories, it seems that we have some good ones for spectra, and some good ones for spaces, but I'm not clear how to go between them. I also want to do all of this in the presence of a monoidal structure which respects everything else, maybe closed, etc. etc.

Any references or guidelines would be dearly appreciated.

-Jon

Fracture Squares of Bousfield Localizations of Spectra

2012-03-12T21:16:48Z

Suppose I have a spectrum $X$ and two homology theories $E$ and $F$. If I look at the Bousfield localizations, $L_E$, $L_F$, $L_{E\vee F}$ and $L_{E\wedge F}$, do I have a homotopy pullback square whose top row is $L_{E\vee F}(X)\to L_E(X)$, and whose lower row is $L_F(X)\to L_{E\wedge F}(X)$? If not, is it known what conditions I need to place on $E$ and $F$ to make this all work out? Does anyone know if I can iterate this process over some set of homology theories?

I went ahead and made this a reference request, because I imagine it could a rather significant answer.

Absence of Maps Between p-local and q-local spectra

2012-01-31T21:16:55Z

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 $[X,Y]_\ast=0$?

Thanks!

Bousfield Complements of Interesting Spectra

2012-10-21T22:22:49Z

For a spectrum $X$, Bousfield constructs a spectrum (which is only well-defined up to Bousfield equivalence) $aX$, which he shows satisfies some nice properties, like $\langle a^2X\rangle=\langle X\rangle$, and converts arbitrary joins to meets and arbitrary meets to joins. If we restrict ourselves to Bousfield idempotent spectra ($\mathbf{DL}$), or even the sensibly named "complemented" spectra ($\mathbf{BA}$), these properties get even better.

For nice spectra, like $K(n)$, $BP$, Thom spectra, finite telescopes, etc., have explicit models of these spectra been thought about or are they of any interest? It seems, though I have not yet read Bousfield's original work on it, that $aX$ is a choice of generator for the class of $X$-acyclics. I am curious to know, though it may not be of any practical use, what the homotopy of $aK(n)$ might look like. We know that $K(n)$ is complemented, so we must have $\langle aK(n)\vee K(n)\rangle=\langle S\rangle$, so clearly $\langle aK(n)\rangle>\langle\bigvee_{m\neq n}K(n)\rangle$.

I guess however that it's actually quite clear what $aT(n)$ (the finite telescope) should be.

Is the stable homotopy category idempotent complete?

2012-10-20T19:09:57Z

Is the stable homotopy category idempotent complete? I have not been able to prove it, and the proof for abelian groups seems to strongly rely on looking at elements.

Thanks, Jon

Answer by Jon Beardsley for Suggestions for good notation

2012-10-15T19:32:35Z

All of the notations created to simplify writing category theory. For instance, the idea of drawing a circular arrow inside of a diagram to indicate that that diagram is commutative. As well as the idea of putting an angle in the top left or bottom right of a square diagram to indicate that it is a pushout or pullback. And finally, the notation of augmenting any of these notations with $\simeq$ to indicate that the diagram is only "up to homotopy".

D(R) versus Ho(HR)?

2012-10-12T20:56:29Z

Given an algebraic ring, how is its derived category related to the homotopy category of HR modules?

Thanks. This is essentially a reference request, since I know there may be a lot (or nothing) to say about this.

Well-Generated Localized Triangulated Categories

2012-10-02T21:57:10Z

Suppose given a well-generated triangulated category with a compatible symmetric monoidal structure, $\mathcal{T}$ (in the sense of Neeman). Is it clear that the image of a localization functor will also be well-generated? If not in true in general, is it easy to prove with respect to the $X$-acyclics (i.e. objects which tensor with $X$ to zero) for some element $X$?

The main thing I'm worried about I suppose (though the second axiom for well-generated-ness is not obvious to me either) is the requirement for the generators to be $\alpha$-small for some cardinal $\alpha$. For instance, the $MU$-localization of the stable homotopy category has no small objects. But perhaps they are $\alpha$-small for some $\alpha$? My understanding of such issues is still somewhat superficial.

Coherent MU_*-Modules

2012-09-23T01:34:11Z

It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega_\ast^U(X)$, is a coherent module over $\Omega_\ast^U$ if $\Omega_\ast^U(X)$ has projective dimension 0 or 1 over $\Omega_\ast^U$. It is stated that in a series of lecture notes by Larry Smith that this result can probably be extended to other complexes (and spectra...). However, these lecture notes are from 1970. Does anyone know if this result has been fully generalized? I guess I mean, is it known precisely how far this result can be extended?

Thanks!

Whitehead Theorem for Harmonic Spectra

2012-09-18T16:58:43Z

What are the chances that, for an arbitrary $p$-local harmonic spectrum $X$, if $K(n)\wedge X\simeq\ast$ for all $n$, then $X$ is contractible? This, I believe, holds for suspension spectra and finite spectra. Does anyone know of any harmonic (i.e. local to $\vee_{n\in\mathbb{N}}K(n)$) spectra for which this does not hold?

Thanks!

Internal Homs in Infinity Categories

2012-09-11T20:47:11Z

Given an $\infty$-category $\mathcal{C}$ with whatever descriptors you wish, how do higher internal homs work? Specifically, given two $n$-morphisms, $f$ and $g$, is it possible (I don't see why not a priori) to have the collection of $n+1$-morphisms between $f$ and $g$ itself be an $n$-morphism? Or, is it more sensible to have this collection be an object of the category? Are there any kind of known paradoxes or anything if we allow one or the other? Even more ridiculously, can we allow that collection to be... I don't know, an $m$-morphism for some $m

Thanks, Jon

Bousfield Lattices for which Minimal Objects Coproduct to Sphere Object

2012-09-04T20:03:05Z

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 class? That is, more specifically, in which cases can we say that Proposition 3.7.5 of Hovey, Palmieri and Strickland's "Axiomatic Stable Homotopy Theory" applies?

Formal group laws arising from localizations of MU

2012-07-06T14:00:07Z

This is sort of a two part question:

1) In one construction of $BP$, the Brown-Peterson spectrum, one uses this idempotent map of Quillen's, $g:MU_{(p)}\to MU_{(p)}$ and from what I can tell, the image of this map is $BP$. I have not worked through the details of this carefully. I guess the main idea is just using Brown representability on the cohomology theory $g^\ast(MU^\ast_{(p)}(-))$. Anyway, what happens if, instead of looking at Quillen's idempotent map, we just look at the localization map $MU\to MU_{(p)}$? Would the cohomology theory and formal group law thereby produced give us the same information as $BP$ but just in a much more unwieldy form?

2) In general, what effect does localization (at a general homology theory) have on complex orientability, and are there interesting cases in which such orientability is preserved and produces interesting formal groups?

Invariant Ideals in Split Hopf Algebroids

2012-07-03T18:04:44Z

Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following:

An ideal $J\subset S$ is invariant under the action of the group $\mathrm{Hom}(B,K)$ if and only if $\eta_R(J)\subset J\Sigma$, where $\eta_R$ is the right unit.

Is it clear where the group action of $\mathrm{Hom}(B,K)$ on $S$ or $J$ comes from? Obviously, by the fact that the algebroid is split, there is an action of $\mathrm{Hom}(B,K)$ on $\mathrm{Hom}(S,K)$.

Comment by Jon Beardsley

2013-05-14T23:15:11Z

That is to say, the fact that it's not (at least around me) first talked about in that way seems to be historical.

Comment by Jon Beardsley

2013-05-14T23:14:45Z

If you're interested in AG, you may be interested to know that descent data is in fact controlled by a coalgebra and its comodules. I'm hardly an authority on the subject, but this seems to be purely a historical accident and not necessarily the "natural" way of things.

Comment by Jon Beardsley

2013-05-02T02:06:54Z

I don't think rudeness is really necessary. One of the things that I really like about this site is that one can ask questions (about things that might be frighteningly "obvious" to others) without being verbally cut-down. Though this may the norm in certain realms of academia, there's no reason for it to be.

Comment by Jon Beardsley

2013-04-05T22:35:05Z

sorry didn't mean for this to get bumped up to the top... not a very interesting question.

Comment by Jon Beardsley

2013-03-01T19:50:33Z

Thanks everyone for your answers. I understand this is an elementary question and perhaps MO was not an appropriate place to bring it up. But there seem to be a lot of simple facts in algebraic topology that are known about spaces and that people feel are obviously true about spectra, but struggle to completely and fully "prove" with all the relevant diagrams when pressed for it. Perhaps this is not one of those things, I don't know. As always, from a humble neophyte, thanks.

Comment by Jon Beardsley

2013-02-19T03:15:56Z

I think there might be a difference in our pagination. The paper I'm looking at by that title only has 31 pages.

Comment by Jon Beardsley

2013-02-19T03:13:06Z

Thanks very much Peter. I've been perusing that paper a little, thinking that the solution to my problem probably lies in there somewhere. I will look specifically at the pages your recommend.

Comment by Jon Beardsley

2013-02-13T03:22:35Z

Is there a category of games? Can I stabilize it? Let's just go ahead and stabilize it. Get to work.

Comment by Jon Beardsley

2013-02-12T20:51:03Z

And yeah, thanks Mark. I'm trying to figure out how that's related.

Comment by Jon Beardsley

2013-02-12T20:49:43Z

Sorry, ABGHR is the paper: arxiv.org/abs/0810.4535

Comment by Jon Beardsley

2013-02-07T17:10:05Z

Perhaps more interesting would be investigating a function whose value at $n$ is "the number of interesting examples of $n$-chotomies in mathematics" as I suspect it would decrease rather quickly. However, philosophically, there is a probably a good reason that there are many examples of dichotomies and trichotomies, as opposed to, say 11-chotomies. Perhaps this is connected to the relative smallness of the human mind, in some sense. Additionally though, such a philosophical argument would show that such a question is pretty silly.

Comment by Jon Beardsley

2013-02-02T19:09:19Z

Yeah that's a good point. I didn't really think about it, and I'm actually not really familiar enough with the standard literature to even realize someone might be confused! But thanks, yeah, I guess basically we can just write it down as the best homotopical-ization of the cotensor of comodule algebras over a (homotopical) Hopf-Algebra. I suppose that's more or less exactly what it is, and Hess shows that in nice cases, this can indeed be made homotopical.

Comment by Jon Beardsley

2013-02-01T23:31:38Z

Sorry, deleted some comments I had made that were mistaken. And thanks Lennart, yeah, I am specifically referring to the sort of homotopy coinvariants discussed in Kathryn Hess' paper arxiv.org/pdf/0902.3393.pdf

Comment by Jon Beardsley

2013-01-18T18:36:18Z

@Urs, haha, no it's great. People are too serious on here anyway.

Comment by Jon Beardsley

2013-01-15T23:50:54Z

@Urs, you're not even talking about math and I'm still confused.