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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 \...
Tim Campion's user avatar
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5 votes
0 answers
279 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&...
Doron Grossman-Naples's user avatar
7 votes
1 answer
367 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: ...
Tim Campion's user avatar
  • 62.2k
9 votes
1 answer
282 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: ...
Tim Campion's user avatar
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1 vote
0 answers
147 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 ...
Miso's user avatar
  • 71
6 votes
1 answer
240 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 ...
Tim Campion's user avatar
  • 62.2k
2 votes
1 answer
123 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? (...
happymath's user avatar
  • 167
3 votes
0 answers
69 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 $\dim H_\ast(X,\mathbb F_p)$....
Tim Campion's user avatar
  • 62.2k
11 votes
1 answer
369 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 ...
Tim Campion's user avatar
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4 votes
0 answers
132 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 \...
Tim Campion's user avatar
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5 votes
0 answers
426 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 $\mathbb{F}_{p^n}$ ...
Doron Grossman-Naples's user avatar
9 votes
1 answer
337 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 ...
kiran's user avatar
  • 2,002
3 votes
1 answer
165 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 ...
Tim Campion's user avatar
  • 62.2k
5 votes
1 answer
280 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 $...
Tim Campion's user avatar
  • 62.2k
25 votes
2 answers
3k 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 $\mathbb{S}$? ...
XT Chen's user avatar
  • 1,094
4 votes
1 answer
223 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) ...
Max's user avatar
  • 95
8 votes
0 answers
355 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 ...
Doron Grossman-Naples's user avatar
5 votes
0 answers
125 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 ...
Tim Campion's user avatar
  • 62.2k
6 votes
1 answer
325 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 ...
Doron Grossman-Naples's user avatar
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(...
Tim Campion's user avatar
  • 62.2k
4 votes
0 answers
150 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 ...
Tim Campion's user avatar
  • 62.2k
4 votes
0 answers
253 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 ...
Doron Grossman-Naples's user avatar
12 votes
1 answer
353 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 ...
kiran's user avatar
  • 2,002
2 votes
0 answers
81 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 ...
taf's user avatar
  • 448
4 votes
1 answer
182 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 ...
kiran's user avatar
  • 2,002
8 votes
1 answer
616 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 $E^{\ast}(BU(n))$...
user avatar
11 votes
1 answer
648 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 ...
Sofía Marlasca Aparicio's user avatar
1 vote
1 answer
130 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 &...
Tim Campion's user avatar
  • 62.2k
3 votes
0 answers
107 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
718 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 ...
Tim Campion's user avatar
  • 62.2k
8 votes
0 answers
228 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 ...
Niall Taggart's user avatar
12 votes
1 answer
273 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 ...
kiran's user avatar
  • 2,002
26 votes
1 answer
813 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 ...
kiran's user avatar
  • 2,002
13 votes
1 answer
479 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 ...
Nicholas Kuhn's user avatar
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 ...
Jack Davies's user avatar
6 votes
1 answer
258 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 ...
N.B.'s user avatar
  • 767
13 votes
3 answers
1k 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 ...
user267839's user avatar
  • 5,938
9 votes
1 answer
434 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 ...
Surojit Ghosh's user avatar
11 votes
0 answers
815 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 ...
Akhil Mathew's user avatar
  • 25.4k
7 votes
2 answers
518 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\}...
Tim Campion's user avatar
  • 62.2k
5 votes
0 answers
167 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 ...
Tim Campion's user avatar
  • 62.2k
18 votes
1 answer
776 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 ...
Nicholas Kuhn's user avatar
6 votes
1 answer
405 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 ...
happymath's user avatar
  • 167
11 votes
1 answer
613 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 ...
Patriot's user avatar
  • 1,038
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 ...
davik's user avatar
  • 2,035
11 votes
0 answers
508 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 ...
wonderich's user avatar
  • 10.4k
10 votes
1 answer
282 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/...
Raghav's user avatar
  • 113
6 votes
0 answers
169 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 ...
Tim Campion's user avatar
  • 62.2k
14 votes
0 answers
306 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: $$X(m, \infty) \to X \to X[0, m].$$ ...
Eric Peterson's user avatar
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)?