Questions tagged [chromatic-homotopy]
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86 questions
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How can a map have negative Adams filtration in the Adams spectral sequence?
Consider the generalized Moore spectra $M(1) = \mathbb{S}/(2)$ and $M(1,4) = \mathbb{S}/(2,v_1^4)$, everything at the prime $p=2$. In [1] the authors construct a $v_2^{32}$-self map on $M(1,4)$. Their ...
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A direct construction of the K(1)-local Atiyah-Bott-Shapiro orientation of KO
Throughout, I fix a prime $p$ (odd, if necessary), and I let $\mathrm{KO}_p$ be $p$-adically completed real $K$-theory.
From Section 7 of Ando-Hopkins-Rezk, we know that the space of $E_\infty$-string ...
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Brown-Comenetz type dual as a target for discrete invertible field theories?
Freed and Hopkins use the spectrum $I\mathbb C^\times$ to define the target of discrete invertible field theories. I have trouble understanding, how this arises. Naively, physically, I would think ...
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Direct way to exhibit $\Sigma^\infty B^h C_p$ as $T(h)$-locally compact?
Note that $L_{T(h-1)} \Sigma^\infty B^h C_p =0$. I think it follows (maybe using dualizability) that $L_{T(h)} \Sigma^\infty B^h C_p$ is $T(h)$-locally compact. (the point is that the compact objects ...
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Counterexample to $\Phi$-good spaces
I am interested in a counterexample of the following type.
I first fix some notation: Let $n$ be a natural and $\Phi$ be the Bousfield-Kuhn functor from based homotopy types to $T_n$-local spectra.
We ...
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A possible refinement of Landweber's filtration theorem
I would like to ask about a possible refinement of Landweber's filtration theorem. In the notation below, $X$ is a finite complex, $BP_*(X)$ is its Brown Peterson homology and $I_n$ are the standard ...
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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 ...
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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 ...
- 3,426
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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 \...
- 68.1k
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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: ...
- 68.1k
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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: ...
- 68.1k
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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 ...
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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 ...
- 68.1k
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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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