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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questions
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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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21 views
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 ...
3
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1answer
40 views
$\sigma (xy) \subseteq \sigma (x)\sigma (y)$ 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 ...
2
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41 views
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 ...
2
votes
1answer
36 views
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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46 views
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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1answer
49 views
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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35 views
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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1answer
33 views
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 $$H_*(E;G)=\mathrm{...
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36 views
Symmetric spectrum associated to a ring
In this paper by Dwyer, Greenlees and Iyengar, we are introduced to symmetric spectra, and more particularly to the notion of an $\mathbb{S}$-algebra.
In both the Notation and terminology ($1.5$), ...
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28 views
Laurent series ring as ${\lim}^1$
Let $R_*$ be a graded ring concentrated in even degrees. I was presented a construction of $R_*((x))$, the ring of Laurent series in the variable $x$ with degree $-d$, as follows. For every $i \in \...
2
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1answer
69 views
Spectrum of the operator $T \in \mathcal{L}(L^2(\Bbb{R}_+))$ defined by $(Tf)(x)=(1āe^{āx})f(x)$
I'm preparing a mathematical physics exam (the last one) and one of the topic is the spectral theory on operators.
I am totally confused on some concept, so please pardon me.
I am studying this ...
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1answer
43 views
Homology of a spectrum
Let $X$ be a spectrum and $E$ another spectrum (it'll be our coefficients, if it makes things easier I'm ok woth assuming $E=H\mathbb Z$)
The definition of $E_nX$ is usually given as $\pi_n(E\wedge X)...
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vote
1answer
72 views
Spectral sequence for homotopy (co)limits
In the accepted answer to this question, user Cary states "What made this spectral sequence tick is that homology/cohomology takes a cofiber sequence to a long exact sequence.".
However this doesn't ...
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1answer
46 views
Is a complex number $\lambda$ in the resolvent set $\rho(T)$ of $T$?
Let $\lambda \notin \{0_{\mathbb C}\} \cup \sigma_p(T)$. Show that $\lambda$ is in the resolvent set $\rho(T)$ of $T \in \mathcal L(\ell^2(\mathbb C))$, where
$$
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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60 views
$KO_*$ 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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27 views
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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1answer
27 views
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 ...
2
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1answer
50 views
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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66 views
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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107 views
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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67 views
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 ...
2
votes
1answer
37 views
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 $\...
3
votes
0answers
84 views
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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0answers
61 views
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?
3
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0answers
155 views
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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70 views
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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60 views
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.
...
4
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1answer
151 views
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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96 views
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^...
4
votes
1answer
110 views
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-...
4
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1answer
124 views
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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36 views
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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197 views
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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21 views
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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176 views
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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46 views
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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1answer
94 views
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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70 views
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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19 views
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 $G$āCW complexes. Is there some analogue result to ...
3
votes
1answer
299 views
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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votes
1answer
222 views
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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49 views
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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1answer
99 views
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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1answer
178 views
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 ...
2
votes
1answer
115 views
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 ...
