All Questions
Tagged with chromatic-homotopy stable-homotopy
39 questions
6
votes
0
answers
357
views
On the nilpotence of the attaching maps for $\mathbb C \mathbb P^\infty$
Consider the usual cell structure on $\mathbb C \mathbb P^\infty$. The skeleta are the $\mathbb C \mathbb P^n$’s, and there is one cell in each even degree. So we have cofiber sequences $S^{2n+1} \to \...
- 64k
7
votes
1
answer
425
views
Does there exist a Bousfield localization of the category of spectra which makes the sphere unbounded below?
Let $Sp$ be the category of spectra. Let $L : Sp \to Sp_L$ be the localization functor onto a reflective subcategory.
Question 1: Is it ever the case that $L(S^0)$ is not bounded below?
Question 2: ...
- 64k
6
votes
1
answer
255
views
Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?
It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...
- 64k
5
votes
0
answers
525
views
Is there anything special about the Honda formal group?
The "standard" Morava E-theory $E_n$ (at a prime $p$) is typically defined using the so-called "Honda formal group law", the unique FGL $\Gamma_n$ over $\mathbb{F}_{p^n}$ ...
- 1,969
3
votes
1
answer
167
views
Can the Picard-graded homotopy of a nonzero object be nilpotent?
Let $\mathcal C$ be a symmetric monoidal stable category such that the thick subcategory generated by the unit is all of $\mathcal C$ -- in particular, every object is dualizable (I'm particularly ...
- 64k
9
votes
0
answers
405
views
What is the Balmer spectrum of the p-complete stable homotopy category?
When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...
- 1,969
4
votes
0
answers
153
views
Are telescopes Noetherian?
Let $p$ be a prime and $h \in \mathbb N$ a height.
Question 1: Does there exist a compact $T(h)$-local spectrum $A$ with a unital multiplication making $\pi_\ast A$ a Noetherian ring?
A priori it's ...
- 64k
8
votes
1
answer
684
views
$E$-(co)homology of $BU(n)$ (Reference request)
I am currently reading Lurie's notes on Chromatic Homotopy Theory (252x) and in Lecture 4 (https://www.math.ias.edu/~lurie/252xnotes/Lecture4.pdf), he skims through the calculation of $E^{\ast}(BU(n))$...
user340793
1
vote
1
answer
136
views
Can a finite, type $n+k$ spectrum be a (non-iterated) colimit of finite, type $n$ spectra for $k \geq 2$?
By the thick subcategory theorem, if $X, Y$ are finite $p$-local spectra of type $m,n$ respectively, then $Y$ can be built from $Y$ in a finite number of "steps" iff $n \geq m$. Here, a &...
- 64k
3
votes
0
answers
109
views
Does $K(n)$ detect minimal $K(n)$-local cell structures?
Let $X$ be a finite spectrum, and let $N = dim_{\mathbb F_p} H_\ast(X;\mathbb F_p)$. I believe that $p$-completion $X^\wedge_p$ may be built as an $N$-cell complex where the cells are shifts of the $p$...
17
votes
2
answers
725
views
For which $n$ does there exist a closed manifold of (chromatic) type $n$?
Let $p$ be a prime and $n \in \mathbb N$. Does there exist a closed manifold which is of type $n$ after $p$-localization?
When $n= 0$ the answer is yes. When $p = 2$ and $n = 1$ we can take $\mathbb R ...
- 64k
26
votes
1
answer
832
views
Are complex-oriented ring spectra determined by their formal group law?
To every complex-oriented ring spectrum $E$ there is associated a formal group law, which is a power series $F_E(x,y)\in E_*[[x,y]]$.
Suppose $E$ and $F$ are two complex-oriented ring spectra and ...
- 2,052
21
votes
1
answer
2k
views
Why does elliptic cohomology fail to be unique up to contractible choice?
It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some ...
- 423
14
votes
3
answers
2k
views
What is so 'coloured' on Chromatic Homotopy Theory
As the title suggest, I would like know the motivation/ historical background
why chromatic homotopy theory is called 'chromatic'. Literally, what
analogy to colors it might have.
Accordings to
...
- 5,966
12
votes
0
answers
879
views
Chromatic blueshift and Tate cohomology
Let $R$ be an $L_n$-local ring spectrum. Then one knows that the Tate construction $R^{tC_p}$ (with respect to the trivial $C_p$-action on $R$) is $L_{n-1}$-local; this "blueshift" result is ...
- 25.6k
11
votes
1
answer
636
views
On the relation between categorification and chromatic redshift
In the introduction to the paper Higher traces, noncommutative motives, and the categorified Chern character, Hoyois, Scherotzke and Sibilla write the following.
An important insight emerging from ...
- 1,098
11
votes
0
answers
533
views
Chromatic Homotopy Theory and Physics
Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
- 10.5k
15
votes
0
answers
313
views
Does virtual Morava K-theory have an Eilenberg-Moore spectral sequence?
In a recent question, Tim Campion was interested in analyzing the Morava $K$–theory of a space $X$ by dissecting the space into connective and coconnective parts: $$X(m, \infty) \to X \to X[0, m].$$ ...
- 6,308
22
votes
2
answers
6k
views
References and resources for (learning) chromatic homotopy theory and related areas
What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?
23
votes
2
answers
2k
views
Latest results in chromatic homotopy theory
I started a PhD in chromatic homotopy three years ago, but I had to quit it due to personal reasons after one year. Last week I was looking at all my notes from that period and I was wondering where ...
- 899
7
votes
1
answer
336
views
Crafting Suspension Spectra
There is a theorem by Hopkins and Smith which states that for every $n > 0$ there is an ideal $I_n = (v_0^{k_0}, \dots, v_n^{k_n})$ such that there exist a spectrum $X_n$ with the following ...
- 899
5
votes
1
answer
202
views
Homology of a limit of spectra + Cofiber
I have a countable sequence of finite suspension spectra $X_i$, whose $BP$-homology is a $BP_*(BP)$-comodule. Let's assume $BP_*(X_i) = \Sigma^{d_i} BP_* / (v_0^{k_0}, \dots v_i^{k_i}),$ for some $d_n$...
- 899
5
votes
1
answer
286
views
Map between homology of spectra
Let $X$ be a suspension spectra whose $BP$-homology is infinitely generated
($BP_*(X) = \Sigma^d BP_*/I$, where $I$ has the form $I=(v_0^{i_0}, \dots , v_n^{i_n})$ such that the homology is a $BP_*(BP)...
- 899
3
votes
1
answer
267
views
Studying the limit of a sequence of spectra knowing their BP-Homology
QUESTION EDITED: There was a mistake, the spectrum i had written before didn't even exist, so a big thanks to the people who made me notice that in the comments.
Let $X_n$ be the spectrum such that $...
- 899
9
votes
0
answers
152
views
How to show that a spectrum X is not Chromatically Complete
There are some criteria which tell us when a spectrum $X$ is chromatically complete (it's the homotopy limit of its chromatic tower):
It has to be p-local and finite, according to the chromatic ...
- 899
11
votes
0
answers
206
views
What are examples of spectra whose mod 2 cohomology contain A//A(n)?
Let $//$ denote the Hopf algebra quotient. We know that:
$$HF_{2}^*(ko) \simeq A//A(1)$$
$$HF_2^*(tmf) \simeq A//A(2)$$
By Hopf invariant one, we know there is no $X$ such that $HF_2^*(X) \simeq A//...
- 3,539
9
votes
0
answers
228
views
Chromatic Completion of Suspension Spectra and affine results
There is the Chromatic Convergence Theorem by Hopkins and Ravanel which states that the homotopy inverse limit of the chromatic tower of a finite spectra $X$ is $X$.
Let's call any spectra with this ...
- 899
28
votes
2
answers
2k
views
Has anyone seen a nice map of multiplicative cohomology theories?
I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere.
I could try to do it myself but I really lack expertise, hence am afraid to miss something or ...
- 18.4k
5
votes
0
answers
216
views
Have chromatic techniques actually been used to compute more stable homotopy groups of spheres?
I have heard the perspective that chromatic homotopy theory is meant to break apart the stable homotopy groups of spheres into manageable pieces, and that this perspective has led to various insights. ...
- 71
63
votes
5
answers
18k
views
What is modern algebraic topology(homotopy theory) about?
At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
4
votes
1
answer
361
views
Localization at the Johnson-Wilson spectrum and rationalization
Is there a clean proof that the $L_n$, localization at $E(n)$, is simply rationalization (i.e. $L_0$) on Eilenberg-MacLane spectra? Eric Peterson asked this here, but I haven't seen an answer.
user62675
6
votes
1
answer
839
views
Uniqueness of Complex Orientation of Morava K-theory
It is known that the $n^{\text{th}}$ Morava $K$-theory at a prime $p$, denoted $K(n)$, is complex oriented. In other words, it admits a theory of Chern classes, or equivalently a morphism of homotopy ...
- 10.5k
26
votes
0
answers
642
views
Chromatic Spectra and Cobordism
I apologize in advance, if some of the things I've written are incorrect.
The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
- 865
4
votes
1
answer
282
views
Is there a geometric interpretation of Johnson-Wilson E(n) analogous to vector bundles for K-theory?
I am reading Ravenel's Localization with Respect to Certain Periodic Homology Theories where he states;
For $n\ge2$, the spectra E(n)
represent periodic homology theories which at present have ...
11
votes
0
answers
648
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 ...
- 10.5k
4
votes
0
answers
224
views
Formal n-buds from BU(n) rather than SU(n)
It's known, from Ravenel's green book, as well as other sources, that we build formal group laws over a ring from n-buds, where an n-bud is essentially a truncated formal group law (sometimes called a ...
- 10.5k
13
votes
1
answer
1k
views
Connection of X(n) spectra to formal group laws
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 ...
- 10.5k
2
votes
1
answer
249
views
Compact MU or BP Modules
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?
- 10.5k
1
vote
1
answer
218
views
Properties of endmorphism rings of E(n),K(n)-localized spheres
Is it known whether or not the endomorphism rings (or ring spectra) of the localized sphere spectra in $L_nSp$ and $L_{K(n)}Sp$ are Noetherian or not? Are they well understood? Perhaps, in the vein of ...
- 10.5k