Questions tagged [spectra]

Use this tag for questions concerning the notion of spectra in algebraic topology. Spectra are natural objects which arises when dealing with stable properties. For spectra of linear operators, please use the tag (spectral-theory) instead.

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Why do these complements of closed sets induced by annihilators form an open cover of the spectrum?

For an important characterization of morphisms to projective spaces, one step in "The Geometry of Schemes" by Eisenbud and Harris is to understand morphisms from affine schemes to projective ...
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Transforming generalized eigenvalue problems

I'm working on generalized eigenvalue problems (GEPs) which is to find the so called eigenvalue $\lambda$ such that $Av-\lambda Bv=0$ some $v \neq 0$. Denote the spectrum of a GEP associated with A ...
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Which (co)homology theories have (co)chain complexes, spectrum edition?

What homological functors $\mathbf{Sp}\rightarrow\mathbf{Ab}$ arise in the form $H_0\circ T$, for $T:\mathbf{Sp}\rightarrow D(\mathbf{Ab})$ a triangulated functor, and dually for cohomological ...
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$\sigma (xy) \subseteq \sigma (x)\sigma (y)$ in Banach algebra.

Let $A$ be a Banach algebra(with unit $e$), $x,y \in A$ and $xy=yx$. Prove that $\sigma(xy) \subseteq \sigma (x) \sigma (y)$, also $\sigma (x+y)\subseteq \sigma(x)+\sigma(y)$. Where $\sigma$ means the ...
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How can I cluster IR spectra (essentially clustering lines)?

This is sorta a cross-post from my post in StackOverflow. Although, rather than seeking help with the code I'm here asking for help with a chemical/mathematical problem associated with the code. I'm ...
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Mahowald-Hopkins theorem

I am trying to recover fully a proof of a result which is attributed to M. Hopkins in the litterature, namely the following: Let $h:S^1\to BGL_1(\mathbb S_p)$ detect the element $1-p$, then the Thom ...
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Why is the Moore spectrum $S\mathbf{Z}_{(J)}$ a ring spectrum?

Let $J$ be a set of primes and consider the Moore spectrum $S\mathbf{Z}_{(J)}$. In his paper 'The localization of spectra with respect to homology', Bousfield writes that $S\mathbf{Z}_{(J)}$ is a ring ...
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A non modern theory of generalized Thom spectra

I've already posted this question on mathoverflow a few days ago and I had no reaction, I hope it is not illegal to repost it on math.stackexchange. I'm new in the subject of stable homotopy theory, ...
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Equivalence between lax monoidal functors and monoids in the functor category.

I'm trying to go through the details of Proposition 3.4 of: https://ncatlab.org/nlab/show/Day+convolution For whatever reason, I don't see how to translate the conditions of a lax monoidal functor ...
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The infinite loop space associated to the spectrum associated to a special $\Gamma$-space

I am currently reading M. Mandel's paper `An Inverse $K$-theory Functor' to extract some results I may wish to use in my masters research project, and I am stuck on one particular claim. On the top of ...
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Basic question about the definition of the homology of a spectrum

The general definition of the homology of a spectrum $E$ with coefficients in an abelian group $G$ is $$H_*(E;G):=\pi_*(E\wedge HG)$$ and I always see people using the equality $$H_*(E;G)=\mathrm{...
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Symmetric spectrum associated to a ring

In this paper by Dwyer, Greenlees and Iyengar, we are introduced to symmetric spectra, and more particularly to the notion of an $\mathbb{S}$-algebra. In both the Notation and terminology ($1.5$), ...
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Laurent series ring as ${\lim}^1$

Let $R_*$ be a graded ring concentrated in even degrees. I was presented a construction of $R_*((x))$, the ring of Laurent series in the variable $x$ with degree $-d$, as follows. For every $i \in \...
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Spectrum of the operator $T \in \mathcal{L}(L^2(\Bbb{R}_+))$ defined by $(Tf)(x)=(1−e^{−x})f(x)$

I'm preparing a mathematical physics exam (the last one) and one of the topic is the spectral theory on operators. I am totally confused on some concept, so please pardon me. I am studying this ...
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Homology of a spectrum

Let $X$ be a spectrum and $E$ another spectrum (it'll be our coefficients, if it makes things easier I'm ok woth assuming $E=H\mathbb Z$) The definition of $E_nX$ is usually given as $\pi_n(E\wedge X)...
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Spectral sequence for homotopy (co)limits

In the accepted answer to this question, user Cary states "What made this spectral sequence tick is that homology/cohomology takes a cofiber sequence to a long exact sequence.". However this doesn't ...
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Is a complex number $\lambda$ in the resolvent set $\rho(T)$ of $T$?

Let $\lambda \notin \{0_{\mathbb C}\} \cup \sigma_p(T)$. Show that $\lambda$ is in the resolvent set $\rho(T)$ of $T \in \mathcal L(\ell^2(\mathbb C))$, where $$ T(x_n) = \left(\frac{x_n}{n}\right)\,. ...
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A homotopy equivalence of $X \wedge Y \to Y \wedge X$ to itself.

In the Adams Bluebook, to define a smash product of CW-spectra, the author uses a homotopy equivalence between two smash product of compactly generated pointed spaces from $S^{1} \wedge Y$ to $Y \...
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$KO_*$ groups of $\mathbb{R}P^\infty$, “Snaiths” theorem for $KO$

I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$? The standard tools to compute these Groups in the complex case rest on the requirement for the cohomology theories $E$...
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Spectral properties via determinant function

I searched through literature but could not find any related topic for the question below. I hope some of you may be able to point me to the right direction. Let $X: \mathbb{R}\rightarrow\mathbb{R}^{...
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Field spectra and Eilenberg--MacLane spectra?

Apparently thanks to a theorem of Hopkins and Smith, every field spectrum splits into a wedge of Morava K-theories, where we allow the cases $K(0) = H \mathbb{Q}$ and $K(\infty) = H \mathbb{F}_p$. I ...
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Why is $E^*(X)$ graded commutative?

Context: I'm reading The stable homotopy category by Malkiewich (see his webpage, at Expository writings), and at this point the author hasn't actually introduced spectra, just the properties we would ...
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Functoriality of twisted K-theory

In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until now. I'm trying to use May-Sigurdsson's definition of Twisted $K$-theory, ...
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Cohomology of spectra as free modules over Steenrod Algebra

I'm studying the construction of Adams spectral sequence in Hatcher's "Algebraic Topology" chapter five; I'm in trouble with an "algebraic statement". Let $X$ be a connective CW-spectrum of finite ...
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Tate construction of the Anderson dual of the sphere spectrum

I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by ...
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if $\lambda \in sp(u)$ why $\lambda \in sp(\phi)$ too?

$E$ est un $K-$espace vectoriel de dimension finie $n$ , et u $ \in L(E) $ et $\phi$ une application de $L(E)$ vers $L(E)$ et soit $\phi(v)=u \circ v$ Montrons que si $\lambda \in sp(u)$ alors $\...
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Does trivial cohomology of spectra imply trivial homology?

It is known that for any spectrum $X$, $H\mathbb{Z}^*(X)=0$ implies that $H\mathbb{Z} \wedge X =0$. Also, for the case $HF_p,$ if we consider $ HF_p^*(X) =0.$ This gives $[HF _ p \wedge X , \sum^i ...
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Spectral decomposition of an operator

Given the operator: $$\begin{pmatrix} i & 0 & - 4\\ 0 & - 3i & 0\\ 2 & 0 & - i \end{pmatrix}$$ Now $det(\lambda1-A)=(z+3i)(z^2+9)=(z+3i)^2(z-3i)$, I know a theorem that says ...
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hermitian operators with the same spectra

I want to prove that if two hermitian matrices say $H_1$ and $H_2$ have the same spectra, then one of them is conjugate of the other by some unitary matrix. $$H_1 = U H_2 U^{\dagger}$$ Any idea?
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Structure of module over Eilenberg MacLane spectrum

Let $HR$ be the Eilenberg-Maclane spectrum for a commutative ring $R$ and $M$ be a module over $HR.$ Then I want to prove that $M$ is a product of Eilenberg-Mac Lane spectra. Construction: Let $\...
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properties of the spectrum in predicate logic

I want to proof that the set of spectra is closed under union and intersection. I know that it works somehow by assuming that the languages of the two formulas $\phi_1$ and $\phi_2$ are disjoint and ...
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Loops Infinity of a spectrum

Let $\mathbf{X}$ be an (orthogonal) spectrum (can assume that it's an $\Omega$-spectrum if this helps give a positive answer) and give the category of orthogonal spectra the stable model structure. ...
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Convenient categories in algebraic topology: their importance, and the role topology plays in their construction

Disclaimer. I have stated three questions but I felt that they are so related that they fit within a single post. Context. After reading Hatcher's Algebraic Topology I wanted to learn more about ...
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Construction of $(p)$-local spectrum.

I have a question regarding the construction of the $p$-local spectrum of a given spectrum $X$. I have seen in many papers people defining the dual notion of it, namely the $p$-completion of $X$, $X_p^...
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Definition of the Berkovich spectrum

I am trying to read these notes: http://www-personal.umich.edu/~takumim/Berkovich.pdf Regarding the Berkovich spectrum. In definition [2.24] it says that the spectrum is the set of bounded (non-...
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Realizing the Berkovich affine line as a union of Berkovich spectrums

I am trying to understand what is the relation of the affine Berkovich space to the Berkovich space on an appropriate polynomial ring. A more exact version of the question is as follows: Let $(K,\...
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Relation of $\mathbb{Z}_2$-cohomology and interger cohomology

In the last chapter of May's Concise course, we have an isomorphism of $\mathbb{Z}_2$-cohomology of spectra, induced by a map of spectra. And then May writes, it can be deduced that the induced map on ...
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Homotopy of wedge sum

In the last chapter of his Concise course in algebraic topology, May states (without proof or reference) that for an arbitrary collection $(X_i)_{i\in I}$ of spectra the following hold: $\pi_n(\prod_{...
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Meaning of cocycle on a spectrum?

What is the meaning of the notation $Z^{p}(F,\mathbb{R})$ where $F$ is a spectrum ? I encountered this in a paper (see the line after remark 4.49) that I am reading, but can not understand the meaning....
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A question regarding generalized cohomology and spectra : proof of $E^{\ast}(S)\otimes\mathbb{R} = H^{\ast}(S;\pi_{\ast}E\otimes \mathbb{R})$

I am reading the paper Quadratic functions in geometry, topology and M theory by M.J.Hopkins and I.M.Singer, and in section 4.8 they say : 'Recall that for any compact $S$, and for any cohomology ...
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spectrum theory in generalized cohomology

why is generalized cohomology theory equivalent to spectrum theory ? specifically, why is that any generalized cohomology theory is represented by a spectrum and a spectrum gives a ...
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Reference for spectra theory (in topology)

Is there any good reference for the theory of spectra (in topology)? Thanks a lot for your help!
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Isomorphism in the case of real symmetric matrices

I'm looking for clarification of the following problem: it's known that determining which graphs are uniquely determined by their spectra is in general a very hard problem. But what about more ...
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Reference request: Representability of multiplicative equivariant cohomology theories

Let $G$ be a topological group, say a compact Lie group, and $e^*_G$ a multiplicative $\mathbb Z$-graded $G$-equivariant cohomology theory defined on $G$–CW complexes. Is there some analogue result to ...
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Is there a stable Hurewicz Theorem?

Given a $(n-1)$-connected spectrum $E$ , is the natural morphism ${\pi _k}\left( E \right) \to {\pi _k}\left( {H\mathbb{Z} \wedge E} \right)$ an isomorphism for $k \leq n$? I think yes, but I can't ...
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Finite extension of Noetherian rings [duplicate]

I've got two noetherian rings $A\subset B$ such that $B$ is a finite $A$-module. Now, if I consider the associated map between spectra that given $q \in \operatorname{Spec} B$ consider $q \cap A \in \...
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Unexplained Mathematical Passage in Fourier Series

just as an FYI, this is an exercise from Digital Signal Processing by Proakis & Manolakis 4th Edition. I need to calculate the spectrum X(F) of a signal given by a graph. That part was easy and I ...
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Spectral realization of the natural transformation from bordism to singular homology

There is a geometric version of the definition of singular homology $H_n$ in terms of continuous maps of "pseudomanifolds" ($n$-dimensional simplicial complexes such that every $(n-1)$-simplex is ...
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Reference request: Adjunction of $\Sigma^\infty$ and $\Omega^\infty$ is monoidal

I would like to see that the map of spectra $\Sigma^\infty \mathbb{C}P^\infty_+ \to KU$ is actually a map of ring spectra, where $KU$ denotes the complex K-theory spectrum and the map is given as the ...
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Splitting of a spectrum as a wedge

Suppose that $i:E\to F$ and $r:F\to E$ are maps of spectra ($S^1$-spectra of topological spaces) such that $r\circ i$ is a homotopy equivalence. Can we always show that the spectrum $F$ splits as a ...