Questions tagged [chromatic-homotopy]
The chromatic-homotopy tag has no usage guidance.
80 questions
6
votes
0
answers
141
views
Are the $K(n)$-local $E_n$-Adams spectral sequences isomorphic to the Adams-Novikov spectral sequences?
Let $H$ be a closed subgroup of the Morava stabilizer group $\mathbb G_n$. [Devinatz-Hopkins, Prop. 6.7] identifies the $K(n)$-local $E_n$-Adams spectral sequence for $E_n^{hH}$ as the homotopy fixed ...
- 155
6
votes
1
answer
822
views
Chromatic homotopy + algebraic geometry =?
In Homotopy Theory there is a famous theorem which shows that every cohomology theory satisfying a certain list of axioms is characterized by a formal group law, and that the spectrum associated to ...
- 2,907
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
5
votes
0
answers
349
views
What, precisely, is a stratification of a stack?
I'm currently working on a structure result for a certain spectral moduli problem, and I've been running into the problem of having to define what, precisely, is meant by the term "stratification&...
- 1,969
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
9
votes
1
answer
328
views
What is the center of Morava $K$-theory?
Let $E$ be an $E_1$ ring spectrum. Then I believe the center of $E$ is an $E_2$ ring spectrum over which $E$ is an $E_1$ algebra, given by the endomorphisms of $E$ as a bimodule over itself.
Question: ...
- 64k
1
vote
0
answers
171
views
A question on $BP$ and $E_\infty$ models for ring spectrums
I am a beginner in this field. My question is
(1) Is the existence of $E_\infty$ ring structure not closed under weak equivalence of ring spectra?
(2) If (1) is true, what is the risk of replacing a ...
- 71
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
3
votes
1
answer
148
views
Homotopy groups of $K(n)$-local $E_n$-modules are $L$-complete
Let $E_n$ be the $n$-th Morava $E$-theory and let $K(n)$ denote the $n$-th Morava $K$-theory.
Question: If $M$ is a $K(n)$-local $E_n$-module, then are the homotopy groups $\pi_*(M)$ $L$-complete? (...
- 177
3
votes
0
answers
70
views
Is every finite spectrum $X$ $K(h)$-locally equivalent to a finite spectrum $Y$ with $\dim (K(h)_\ast Y) = \dim ((H\mathbb F_p)_\ast Y)$?
Let $X$ be a finite spectrum and $K = K(h)$ be the $h$th Morava $K$-theory at the prime $p$. Then $\dim_{K_\ast} K_\ast X$ is increasing in $h$, and eventually constant at $\dim H_\ast(X,\mathbb F_p)$....
- 64k
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 64k
4
votes
0
answers
143
views
Which limits commute with all colimits in $T(h)$-local spectra?
Consider the category $Sp_{T(h)}$ of $T(h)$-local spectra. Let $J, K$ be small $\infty$-categories. Recall that $J$-limits said to commute with $K$-colimits in $Sp_{T(h)}$ if, for all functors $F : J \...
- 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
10
votes
1
answer
424
views
Does every complex orientable $E_\infty$-ring admit an $E_\infty$ complex orientation?
A ring spectrum $E$ is complex oriented if it is equipped with a ring map $MU\rightarrow E$. It is complex orientable if such a ring map exists.
An $E_\infty$-ring $E$ is $E_\infty$-complex oriented ...
- 2,052
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
5
votes
1
answer
295
views
If $\pi_\ast A$ is graded-commutative, then is $A_\ast$ a lax monoidal functor?
Let $A$ be a homotopy ring spectrum. Then the homology theory $A_\ast : Spectra \to GrAb$ lifts to a homology theory valued in $GrMod(\pi_\ast A)$. If $A$ is homotopy commutative, then this functor $...
- 64k
27
votes
2
answers
4k
views
Why the sphere spectrum is more correct than $\mathbb{Z}$?
One may argue that $\mathbb{S}$ is more correct than $\mathbb{Z}$. Can anyone make it more explicitly? For example, what information will be lost if we work in $\mathbb{Z}$ instead of $\mathbb{S}$？
...
- 1,168
4
votes
1
answer
239
views
On the sparsity of the descent spectral sequence computing homotopy groups of the K(n)-local sphere
There is a descent spectral sequence computing $\pi_*L_{K(n)}S^0$ with $E_2$-term
$$E_2^{s,t}\cong H^s_c(\mathbb{G}_n,(E_n)_t)$$
It is mentioned in Barthel-Beaudry (in the description of Figure 3.30) ...
- 155
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
5
votes
0
answers
129
views
Solving polynomial equations in $K(h)$-local or $T(h)$-local spectra?
This is the same question as an earlier question of mine, except in a different category.
Let $Spt_{T(h)}^{fin}$ be the category of finite $T(h)$-local spectra. Let $K_0^\oplus(Spt_{T(h)}^{fin})$ be ...
- 64k
6
votes
1
answer
374
views
Does the Lie algebra structure on rational homotopy groups reflect similar information to the formal group structure in characteristic p?
It's well known (c.f. Quillen and Sullivan) that the rational homotopy theory of spaces is equivalent to the homotopy theory of rational DG-algebras; in particular, rational spaces and rational ...
- 1,969
4
votes
2
answers
200
views
Is $\operatorname{dim}_{K(h)_\ast} K(h)_\ast X$ increasing in $h$?
Let $X$ be a finite $p$-local spectrum. For each $h \in \mathbb{N} \cup \{\infty\}$, let $K(h)$ be Morava $K$-theory of height $h$. Recall that the coefficients $K(h)_\ast$ are a graded field, and $K(...
- 64k
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
4
votes
0
answers
274
views
Is there any use for n-dimensional formal group laws in chromatic homotopy?
Chromatic homotopy tends to mainly focus on $1$-dimensional (commutative) FGLs. From a geometric perspective, this is because line bundles form a group and n-plane bundles don't, so the first Chern ...
- 1,969
12
votes
1
answer
360
views
Can the Bousfield class of projective space be computed directly?
Recall that the Bousfield class of a spectrum $E$, written $\langle E\rangle$, is the class of spectra $X$ such that $X\wedge E$ is not contractible. For example the Bousfield class of any of the ...
- 2,052
2
votes
0
answers
83
views
Map to study $K(n)$-local Picard Group
Let $R$ be an $E_{\infty}$-ring. There's a fiber/cofiber sequence $S$: $gl_1 R \to \text{Pic}(R) \to H(\text{Pic}(R))$, where $\text{Pic}(R) =\pi_0 \text{Pic}(R)$ is the Picard group of $R$. Rotating ...
- 448
4
votes
1
answer
192
views
Are Landweber exact spectra determined by their coefficient ring?
Let $E$ be a Landweber exact ring spectrum. That is, we have a map of homotopy ring spectra $MU\rightarrow E$ and an isomorphism of homology theories $E_*X\simeq MU_*X\otimes_{MU_*}E_*$. Is the ...
- 2,052
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
12
votes
1
answer
703
views
Does the spectrum of Morava E-theory depend only on height?
I almost expect the answer to this question to be no, but I can't find it explicitly said anywhere.
Given a formal group law $f$ of height $n$ over a perfect field $k$ of characteristic $p$, we can ...
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
8
votes
0
answers
232
views
Projective $BP_\ast$-dimension of the $BP$-homology of classifying spaces of finite groups
Fix a prime $p$ and let $G$ be a finite group. Do we know the projective dimension of $BP_\ast (BG)$ as a graded $BP_\ast$-module? Or at least that it is finite?
My guess is the following:
The ...
- 901
12
votes
1
answer
283
views
Is every complex oriented ring spectrum with additive FGL an Eilenberg-Maclane spectrum?
Suppose $E$ is a complex-oriented ring spectrum whose formal group law is isomorphic to the additive one. As the title suggests, we might as well change the complex orientation so that the formal ...
- 2,052
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
13
votes
1
answer
493
views
Chromatic orientability of manifolds
If a compact manifold $M$ with empty boundary is oriented with respect to all the connective Morava $K$-theories $k(n)_*$, localized at a prime $p$, can one conclude that $M$ is orientable with ...
- 11.2k
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
6
votes
1
answer
284
views
Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum
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 ...
- 767
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
9
votes
1
answer
459
views
Applications of equivariant homotopy theory in chromatic homotopy theory
I usually do computations in equivariant homotopy theory, but I would like to learn chromatic homotopy theory where one may use the equivariant techniques, e.g., slice spectral sequences, etc.
For ...
- 745
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
7
votes
2
answers
534
views
Chromatic t-structures?
Questions: Fix a prime $p$ and $n \in \mathbb N_{\geq 1}$.
Does the category $Sp_{K(n)}$ of $K(n)$-local spectra admit a nontrivial $t$-structure?
By "nontrivial", I simply mean that $\{0\}...
- 64k
5
votes
0
answers
168
views
chromatic minimal cell structures
If $X$ is a finite $p$-local spectrum, then the minimal number of cells needed to construct $X$ is exactly $\dim_{\mathbb F_p} H_\ast(X,\mathbb F_p)$. Is there an analogous result in the $K(n)$-local ...
- 64k
18
votes
1
answer
840
views
Are the AHSS and Adams spectral sequence the same when computing connective Morava K-theory of a space?
Let $k(n)$ be the $n$th connective Morava K-theory, with $k(n)_* = \mathbb F_p[v]$ where $|v| = 2p^n-2$. If $X$ is a space or a spectrum (assumed bounded below), one can compute $k(n)_*(X)$ using ...
- 11.2k
6
votes
1
answer
412
views
Descent for $K(1)$-local spectra
For odd primes, we have an equalizer diagram for the $K(1)$- local sphere given by
$$L_{K(1)}S \rightarrow K{{ \xrightarrow{\Psi^g}}\atop{\xrightarrow[i_K ] {}}} K$$
where $g$ is a topological ...
- 177
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
8
votes
2
answers
1k
views
Why do we study complex orientable cohomology theories
It seems that much of the literature in stable homotopy theory seems to study complex orientable cohomology theories. What is the reason of restricting to this class of multiplicative cohomology ...
- 2,356
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
10
votes
1
answer
286
views
Formal group law for oriented bordism
From this answer I learned that the coefficient ring $MSO^{*}[1/2]$ of oriented bordism with 2 inverted supports an odd formal group law and is infact the universal such ring. Is there a reference/...
- 113
7
votes
0
answers
172
views
Is there an $\infty$-topos of monochromatic spaces?
Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain ...
- 64k