Questions tagged [spectral-sequences]
The spectral-sequences tag has no usage guidance.
388 questions
2
votes
1
answer
148
views
An attempt at an alternative calculation of the rank of $\pi_n(MO)$
$\newcommand{\a}{\mathfrak a}\newcommand{\Z}{\mathbb Z}$Let $MO$ be the Thom Spectrum, then I am trying to come up with an alternative calculation that the rank of $\pi_n(MO)$ as a $\Z_2$ vector space ...
- 391
2
votes
0
answers
138
views
Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 12.9k
11
votes
1
answer
957
views
Technology for various models of spectra
There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of $S$-...
CommunityBot
- 1
0
votes
0
answers
80
views
Relation between Chow groups and K theory
I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence
$$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
- 3,597
2
votes
1
answer
201
views
Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence
In A Primer on Spectral Sequences (also later published in More Concise Algebraic Topology), J. Peter May describes the Serre Spectral Sequence for any homology theory. To recap, suppose $p\colon E\...
- 9,263
6
votes
0
answers
141
views
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 ...
- 155
4
votes
0
answers
87
views
Lifting maps on the spectral sequence of a double complex to the derived category
Question
The differentials on the $(r+1)$th page of a spectral sequence are maps on the cohomologies of the complexes on $r$th page. So, between two adjacent complexes $K^\bullet,L^\bullet$ on the $r$...
4
votes
1
answer
514
views
A question about spectral sequences
In the following proof (from The pontrjagin numbers of an orbit map and generalized G-signature theorem by Hsu-Tung Ku & Mei-Chin Ku https://link.springer.com/chapter/10.1007/BFb0085610), it is ...
- 1,367
27
votes
0
answers
470
views
Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,020
3
votes
0
answers
90
views
Topological groups satisfying the Borel transgression theorem
I am using the Borel transgression theorem as given in Mimura and Toda's "Topology of Lie groups I and II", page 378, Theorem 2.7. I know that it applies when the fiber has the homotopy type ...
- 63.9k
13
votes
5
answers
2k
views
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live ...
- 101
7
votes
1
answer
845
views
Mayer Vietoris Spectral sequence for topological K theory
In Sheaf theory one can obtain the Mayer Vietoris spectral sequence for cohomology. For $\mathcal{U}$ an open cover of $X$ we get the convergence
$E_2^{pq} = \check H^p(\mathcal{U},H^q(-,F)) \...
- 18.4k
3
votes
1
answer
223
views
A morphism of double complexes induces a qis on total complexes under certain hypotheses. Proof involving a spectral sequence
$\def\Tot{\operatorname{Tot}}
\def\Ker{\operatorname{Ker}}$I am trying to understand the proof of Lemma 0133 of the Stacks Project. Note that hypotheses (3) and (4) can be restated by saying: extend ...
- 2,368
1
vote
0
answers
87
views
The derived exact couple of an exact couple without chasing elements
$\def\Ker{\operatorname{Ker}}
\def\Im{\operatorname{Im}}$Given an exact couple $(A,E,\alpha,f,g)$ in some abelian category, we define its derived exact couple $(A',E',\alpha',f',g')$ to be $A'=\alpha ...
1
vote
0
answers
101
views
Categorification of spectral sequence
All sorts of things are categorified. What about spectral sequences?
Question: What is a categorification of a spectral sequence?
Talking through my hat, I could imagine an $\infty$-category (...
- 12.4k
2
votes
1
answer
191
views
Cohomology class of fiber bundle
Suppose I have a fiber bundle $\pi: E\rightarrow B$, with fiber $F$, such that the Serre spectral sequence on cohomology is immediately degenerate. In other words, $H^*(E)=H^*(B)\otimes H^*(F)$.
I ...
- 39.3k
4
votes
2
answers
290
views
Loop-space functor on cohomology
For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$.
More concretely, $\omega$ is given by the Puppe sequence
$$\...
- 35.9k
1
vote
0
answers
78
views
A question about the localization theorem of Borel-Hsiang and spectral sequence
Suppose that $T$ is a torus acting on a topological space $X
$. Let $T\longrightarrow E_{T}\longrightarrow B_{T}$ be the universal $T$-bundle. Let $X\longrightarrow X_{T}\longrightarrow B_{T}$ be the ...
- 1,367
0
votes
0
answers
47
views
Generalized edge map in spectral sequence of double complex
suppose we have a cohomologically indexed double complex $C^{\bullet,\bullet}$ with its spectral sequence
$$E_2^{p,q}=H^p_{vert}(H^q_{horiz}(C))\Rightarrow H^{p+q}(C)$$
and suppose that the horizontal ...
- 2,054
3
votes
1
answer
421
views
Spectral sequence in Adams's book, Theorem 8.2
I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...
- 16.2k
2
votes
2
answers
126
views
Extensions of $G$-modules parametrized by $H^1$
Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess one can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
- 38.6k
9
votes
1
answer
265
views
Is there a correction to the failure of geometric morphisms to preserve internal homs?
Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
- 3,312
2
votes
0
answers
161
views
Vanishing differential of Brown-Gersten-Quillen spectral sequence
Let $k$ be an algebraically closed field of characteristic zero and $X$ be an affine, simplicial toric 3-fold over the field $k$. I am trying to use the Brown-Gersten-Quillen spectral sequence to ...
- 639
8
votes
1
answer
441
views
Reference request: cohomology of Eilenberg Maclane spaces with $p$-local groups
In Rudyak's 'On Thom Spectra, Orientability, and Cobordism', the following fact is used:
Let $\mathbb{Z}_{(p)}$ be $\mathbb{Z}$ localized at the ideal $(p)$. Let $\pi,\tau$ be two cyclic $\mathbb{...
5
votes
1
answer
472
views
Two spectral sequences arising from a simplicial spectrum
Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization.
Let's assume each $X_n$ is connective.
From this situation, we can form two filtrations on $X$: the ...
- 2,856
2
votes
0
answers
179
views
Differentials in the Atiyah-Hirzebruch spectral sequence for a bounded generalized cohomology theory
$\newcommand{\res}{\mathrm{res}}$Let $X$ be a connected finite CW complex, and let $E$ be a bounded spectrum. For simplicity, let me assume that it has homotopy groups concentrated in degrees 0,1,2.
...
- 201
2
votes
0
answers
102
views
Alternative construction for the loop space (?)
There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...
- 2,152
1
vote
1
answer
513
views
Why should we study the total complex?
Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
- 37.8k
4
votes
2
answers
409
views
Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
- 441
1
vote
0
answers
107
views
Spectral sequence for a truncated semi cosimplicial space
Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
- 9,263
2
votes
0
answers
147
views
Homotopy equivalence of chain complexes from subcomplexes and quotient complexes
Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
- 673
11
votes
2
answers
858
views
Spectral sequences and short exact sequences
Suppose I take a short exact sequence of filtered chain complexes:
$$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$
We assume that $p$ and $q$ are filtration-preserving, so that $p(F_rA)\subseteq ...
- 9,263
4
votes
1
answer
278
views
Collapse of spectral sequence computing Equivariant cohomology
I have already posted this question on math.stackexchanges but I got no answer and I decided to post it here.
Let us consider the fibration
$$
M\hookrightarrow EG\times_{\varphi}M\twoheadrightarrow BG
...
- 52.7k
7
votes
1
answer
613
views
Image of J in the classical Adams Spectral Sequence
Hey all,
I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E_2$ page of ...
- 2,821
5
votes
0
answers
660
views
Hypercohomology spectral sequence from the derived category point of view
Let $F\colon \mathsf{A}\to\mathsf{B}$ be an additive functor between abelian categories and let $M$ be a complex on $\mathsf{A}$. There's a "hypercohomology spectral sequence"
$$E_1^{i,j}=\...
- 773
6
votes
1
answer
375
views
Different flavours of Vassiliev Conjecture
There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...
- 2,152
2
votes
1
answer
215
views
Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler)
Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering
$$ \dotsb \...
CommunityBot
- 1
2
votes
0
answers
284
views
Notation for spectral sequences [closed]
Every single spectral sequence I have seen in my life was denoted by $E$. Even when there is more than one spectral sequence, people tend to use the same letter with some workaround (e.g. a fourth ...
- 2,152
1
vote
1
answer
138
views
Relation between $G_0(X)$ and $\mathrm{Cl}(X)$ for a normal variety $X$
Let $k$ be an algebraically closed field and $X$ be a normal variety over $k$.
I am trying to show that there is a surjective group homomorphism $G_0(X)\rightarrow \mathbb{Z}\oplus \mathrm{Cl}(X)$, by ...
- 63.9k
1
vote
0
answers
86
views
Image of the boundary maps in the homological spectral sequence of a filtration of a chain complex
I'm trying to understand the construction of the homological spectral sequence of a filtration given in C.A.Weibel ''An introduction to homological algebra''. Here, they start with a filtration of a ...
- 911
5
votes
1
answer
924
views
When is the cohomology of a fiber bundle a tensor product?
Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{...
- 7,609
16
votes
1
answer
776
views
The second stable homotopy group
I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\...
- 663
0
votes
1
answer
238
views
Infinite dimensional homology and spectral sequences
I am new to spectral sequences, so I'm not sure about the difficulty of this question.
Suppose we have a filtration of a chain complex $\emptyset=D_{-1}\subset D_{0}\subset D_1\subset\dots D_n= C$ and ...
- 16.2k
6
votes
1
answer
518
views
Leray spectral sequence and pullbacks
I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence:
Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E_2^...
- 35.2k
2
votes
1
answer
238
views
For locally profinite groups $H\lhd G$, is there a spectral sequence $\newcommand\@[2]{{\rm Ext}_#1^{#2}(\pi_1,\pi_2)}H^p(G/H,\@Hq)\implies\@G{p+q}$?
Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi_1$ and $\pi_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows?
$$...
- 769
0
votes
0
answers
185
views
Interpreting the edges in the Serre spectral sequence
Let $F \hookrightarrow E \stackrel{\pi}{\rightarrow} B$ be a fiber bundle. For simplicity, assume that $F$ is connected, that $B$ is $1$-connected, and that $B$ is a CW complex. Consider the Serre ...
1
vote
0
answers
167
views
Spectral sequence for two fibrations
Given maps of fibrations, i.e. commutative diagrams of smooth manifolds
$$\begin{matrix}
\ F & \to & E &\to & B \\\
\downarrow & & \downarrow & & \downarrow \\\
\ F'...
- 103
4
votes
0
answers
170
views
infinite families in stable homotopy groups
The question is about infinite families in stable homotopy groups. Yes, there are some Q&A about the topic.
But I wonder if the order of Mahowald's elements is known?
in Green Book it mentioned ...
- 63.9k
4
votes
0
answers
102
views
"Standard computations" with stable Hopf invariants
I am struggling in understanding the proof of Lemma 10.6 of the paper "Mapping class groups and function spaces" by Bodigheimer, Cohen and Peim http://www.math.uni-bonn.de/people/cfb/...
- 641
7
votes
1
answer
261
views
Relation between cohomology operations and the Adams spectral sequence
$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Cone}{Cone}$
I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...
- 9,263