Questions tagged [chromatic-homotopy]
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78
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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 \...
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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&...
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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: ...
- 62.6k
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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: ...
- 62.6k
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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
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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 ...
- 62.6k
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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? (...
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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 $\dim H_\ast(X,\mathbb F_p)$....
- 62.6k
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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 ...
- 62.6k
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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 \...
- 62.6k
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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 $\mathbb{F}_{p^n}$ ...
- 1,837
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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 ...
- 2,002
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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 ...
- 62.6k
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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 $...
- 62.6k
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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 $\mathbb{S}$？
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