The stable-homotopy tag has no wiki summary.

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### Good morphisms of distinguished triangles: can Neeman's method be applied to the motivic stable homotopy category?

It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some ...

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### The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?

I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.
For the algebraic cobordism theory ...

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### Lambda-operations on stable homotopy groups of spheres

The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable ...

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### Is a wedge of spheres an $E_\infty$ ring spectrum?

The wedge sum $\bigvee_{k \in 2 \mathbb{Z}} S^{k}$ is an $A_\infty$-ring spectrum: the connective cover is the free $A_\infty$-ring on the sphere $S^2$, if I'm not mistaken, and then one inverts the ...

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### Periodicity Theorem for D(R)

For a derived category of a Noetherian ring (or perhaps more generally), can we talk about a Periodicity Theorem? We have Thick Subcategory Theorems and Nilpotence Theorems (HPS 91) for D(R), and in ...

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### Monoidal Model Categories with Suspension Functor

This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model ...

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### On the natural (bigraded) homotopy groups of a simplicial object in a model category

$\def\mc{\mathcal} \def\sm{\wedge}$
This question stems from the Goerss-Hopkins paper Moduli Problems for Structured Ring Spectra. Let me begin by attempting to summarize the relevant framework -- ...

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### Bousfield Complements of Interesting Spectra

For a spectrum $X$, Bousfield constructs a spectrum (which is only well-defined up to Bousfield equivalence) $aX$, which he shows satisfies some nice properties, like $\langle a^2X\rangle=\langle ...

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### K(r)-localization and monochromatic layers in the chromatic spectral sequence

While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.
The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for ...

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### Whitehead Theorem for Harmonic Spectra

What are the chances that, for an arbitrary $p$-local harmonic spectrum $X$, if $K(n)\wedge X\simeq\ast$ for all $n$, then $X$ is contractible? This, I believe, holds for suspension spectra and finite ...

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### Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence.

In Shipley's paper http://arxiv.org/abs/math/0209215 she proves a Quillen equivalence between the category of $H\mathbb Z$-modules and dg $\mathbb Z$-modules. So, to a chain complex $C$, she assigns a ...

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### Bousfield Lattices for which Minimal Objects Coproduct to Sphere Object

Is it known what conditions we require of a stable homotopy category to have $\langle S\rangle = \coprod\limits_{\mathbb{N}}\langle K(n)\rangle$, where $\langle K(n)\rangle$ is some minimal Bousfield ...

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### Image of J in the classical Adams Spectral Sequence

Hey all,
I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E_2$ page of ...

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### Finiteness of stable homotopy groups of spheres

Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or ...

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### Cohomology theory associated to the spectrum BG

Hi, I've recently been interested in Stable Homotopy Theory and was reading this text to understand some basics: http://www.maths.ed.ac.uk/~aar/papers/carlmilg.pdf
Near the end of the text (p582) we ...

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### Reference on the question mark cell complex

The question mark complex is a finite spectrum whose cohomology looks like a "question mark" (when drawn as a module over the Steenrod algebra): that is, there is an element in dimension zero $a_0$, ...

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### Is there an algebro-geometric description of $\nu$?

Motivation: According to the "chromatic" picture of stable homotopy, we should think of the moduli stack $M_{FG}$ of formal groups as a "good approximation" to the stable homotopy category (more ...

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### Is there an obvious reason why p-localization of spectra is a finite localization?

Is there an obvious reason why $p$-localization of spectra is a "finite" localization in the sense of Haynes Miller? In other words, is there an obvious reason why the localizing subcategory (of the ...

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### How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$

Hey everybody,
I think this question might be just a simple oversight on my part, but this has been bugging me a few days.
I am reading Hatcher's Spectral Sequences book, and trying to understand ...

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### Formal group laws arising from localizations of MU

This is sort of a two part question:
1) In one construction of $BP$, the Brown-Peterson spectrum, one uses this idempotent map of Quillen's, $g:MU_{(p)}\to MU_{(p)}$ and from what I can tell, the ...

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### Geometric interpretation of families in the stable homotopy groups of spheres

There are infinite families in the stable homotopy groups of spheres; many of these can be seen by looking for "periodicity" phenomena in the Adams-Novikov spectral sequence. An example is the image ...

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### Second homotopy group of the mod 2 Moore spectrum

Let $S/2$ be the mod 2 Moore spectrum (i.e. the cofiber of $2: S \to S$). Then multiplication by 2 acts nontrivially on this spectrum: the homotopy groups of $S/2$ are all $\mathbb{Z}/4$-modules by a ...

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### What is the first interesting matric Toda bracket in the stable homotopy of the sphere?

Feel free to gloss â€˜interestingâ€™ as you see fit. One way:
1. What is the lowest degree matric Toda bracket in $\pi_\ast(S)$ that doesn't contain zero?
By â€˜degreeâ€™ I mean total homotopical ...

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### How to construct maps between (co)fibre sequences in a stable $\infty$-category?

Fix a stable $\infty$-category $\mathcal{C}$ and two (co)fibre sequences $a \rightarrow b \rightarrow c$ and $x \rightarrow y \rightarrow z$ in $\mathcal{C}$. Now suppose we are given a map $a ...

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### Can a nontrivial spectrum smash to zero with $K$-theory?

Let $E $ be a (possibly nonconnective) spectrum. Suppose $E \wedge K = 0$ (where $K$ is complex $K$-theory). Does it follow that $E = 0$?

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### Refinement of concept of support of a module

My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...

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### Non-Noetherian Stable Homotopy

There seems to be quite a bit of theory developed to deal with "stable homotopy" in the sense of the derived category of a Noetherian ring, or just any situation where the endomorphism ring of the ...

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### 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 ...

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### Is the 4-line of the E_2 term of the classical Adams spectral sequence known?

In other words:
What is $\mathrm{Ext}_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?
If the 4-line is not known, how much is known about it?
Here, $\mathcal{A}$ is the 2-primary Steenrod ...

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### Chromatic convergence of E(n)-localized homotopy categories

Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy ...

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### Gluing Local Spectra

Suppose we have four spectra: $E_1$, $E_2$, $X_1$ and $X_2$ where $X_i$ is $E_i$ local and $L_{E_1\wedge E_2}X_1\simeq L_{E_1\wedge E_2}X_2$. Of course what this really means (I think), if we're ...

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### The Bousfield Class of the Infinite Wedge of Telescopes of Finite Spectra

The spectrum $T(n)$ which is the telescope of a finite spectrum of type n along its self-map, has a unique Bousfield class $\langle T(n)\rangle$ which only depends on $n$. It is also known, from ...

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### Fracture Squares of Bousfield Localizations of Spectra

Suppose I have a spectrum $X$ and two homology theories $E$ and $F$. If I look at the Bousfield localizations, $L_E$, $L_F$, $L_{E\vee F}$ and $L_{E\wedge F}$, do I have a homotopy pullback square ...

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### Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?

The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it ...

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### Relationship of Bousfield Classes of Morava K-theories

Suppose we have $\langle K(n)\rangle$ and $\langle K(n-1) \rangle$ for some fixed prime $p$. do we know whether or not $\langle K(n) \rangle \geq \langle K(n-1) \rangle$ or $\langle K(n-1) \rangle ...

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### Bousfield Classes

This question has a few parts:
1) Is the Bousfield class of $\langle E\rangle$ the class of $E$-acyclics, i.e. $\langle E\rangle=\left\{ X:E\wedge X=0\right\}$ or is it the class of spectra which are ...

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### A Model for the Moore Spectrum of $\mathbb{Z}_{(p)}$

Is it true that the Moore spectrum for the group $\mathbb{Z}_{(p)}$ can be constructed by smashing $\mathbb{S}$ with $q^{-1}\mathbb{S}$ for each $q\neq p$ (here both $q$ and $p$ are primes). It seems ...

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### Absence of Maps Between p-local and q-local spectra

Suppose $X$ and $Y$ are spectra (or homotopy classes thereof) such that $X$ is p-local and $Y$ is q-local, for primes $p\neq q$. Is it indeed true then, and if so how would one show that ...

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### The stable-homotopy-homology-theory

Hi
Is there a way to stabilise relative homotopy groups into giving the stable-homotopy-homology-functor?
The fact that the homotopy excision theorem holds for exactly the same kind of pair that ...

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### Homotopy type of tensors of Moore spectra

I would like to hear what's known about the homotopy type of smash products of mod-$p^j$ Moore spectra, for $p$ an odd prime.
First, here is what I'm specifically interested in: there is a short ...

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### Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem

I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...

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### Formal Group Laws on Ring Spectra?

Given a (graded) ring $R$, to define a formal group law it is equivalent to define a ring homomorphism $\phi:L\to R$ where $L$ is Lazard's ring. Is there any notion of defining a formal group law on ...

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### do spectra have diagonal maps?

Topological spaces have diagonal maps $X \rightarrow X \times X$ and $X \rightarrow X \wedge X$, and suspension spectra also have diagonal maps $\Sigma^\infty X \rightarrow \Sigma^\infty(X \wedge X) ...

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### Dedekind Spectra

Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum? Additionally, is ...

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### What are the best known results for the stable homotopy groups of spheres?

There are a number of proposed ways to compute the stable homotopy groups of spheres. One can rather peculiarly consider stable (co)homotopy of an Eilenberg Maclane spectrum as a generalised ...

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### Modern Source for Spectra (including Ring Spectra)

I am looking for a modern introduction to Spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what has been ...

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### stable homotopy of BG_+?

Let $G$ be an abelian group (if this helps, let's say also finite). Let $BG_+$ be the classifying space together with a disjoint base point. What are the stable homotopy groups
$$\pi^s_m (BG_+) := ...

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### Stable homotopy theory of orbifolds

Is there a notion of stable homotopy, spectrum, or a stable homotopy category which corresponds to orbifolds and orbispaces, in the same way that classical stable homotopy theory corresponds to ...

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### Are $G$-spectra the same as modules over a “group ring spectrum”?

Let $G$ be a finite group (maybe this will also work when $G$ is compact, or something, but to be safe we'll let it be finite). I imagine it's quite natural to ask: is the category of $G$-spectra ...

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### What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
...