Questions tagged [spectral-sequences]
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407 questions
1 vote
0 answers
41 views
Does local flag compatibility over $S$-integers imply $(n-2)$-connectivity?
Let $R$ be a Dedekind domain, e.g. $R=\mathcal{O}_S$, and $n\ge 1$. Write $T(R^n)$ for the simplicial complex whose $r$-simplices are chains of proper, nonzero direct summands of $R^n$ of length $r$ (...
2 votes
1 answer
219 views
Isomorphisms of sheaves of Abelian groups in Hesselholt's "Topological Hochschild Homology and the Hasse-Weil Zeta function"
This is from §5, p. 13, of Hesselholt's "Topological Hochschild Homology and the Hasse-Weil Zeta function". The author claims there is a family of isomorphisms \begin{equation*} \phantom{\...
- 1,923
3 votes
0 answers
234 views
Computing a certain group of homotopy classes of maps
Let, $G/PL \cong Y \times K(Z_2,6) \times K(Z,8) \times \cdots$. Where, $Y$ fits into the fibration sequence $K(Z,4) \to Y \to K(Z_2,2) \xrightarrow{\delta sq^2} K(Z,5) \to \cdots$ and $\delta$ is ...
10 votes
2 answers
559 views
Introduction or Description of Mahowald's Sq1, Sq2 Diagrams
While reading Mahowald's paper on bo-Resolutions There are some very nicely illustrated modules on pg. 373: I have seen this diagrams occasionally but I am curious if any one knows when and who (i.e. ...
- 263
4 votes
0 answers
146 views
Eilenberg–Moore spectral sequence for dgas
Given $M$, $N$ dg-modules over a dg-algebra $A$ which is over a commutative unital ring $R$. There are spectral sequences: $$E_{p,q}^2 = \operatorname{Tor}_{p,q}^{H^*(A)}(H^*(M),H^*(N)) \implies \...
- 41
7 votes
1 answer
432 views
Two cohomology Cartan-Leray spectral sequences?
Let $X$ be a nice enough topological space and $p\colon X^{\prime}\rightarrow X$ a regular covering space with structure group $G$. Then, there is a Cartan-Leray homological spectral sequence $E^2_{p,...
- 1,912
28 votes
6 answers
4k views
Spectral sequences every mathematician should know
Reading mathematical articles, I sometimes see how mathematicians pull out amazing spectral sequences seemingly at will. While many are built using standard techniques like exact couples or filtered ...
3 votes
0 answers
115 views
Using spectral sequences to prove the Mayer Vietoris sequence on local cohomology is exact
In "On Some Local Cohomology Modules" by Lyubeznik, specifically Theorem 2.1, he gives a spectral sequence and writes "It is not hard to see that if n = 2, i.e. there are just two ...
- 183
2 votes
1 answer
289 views
The Serre spectral sequence of the Borel fibration and monodromy action
Let $R$ be a commutative ring with unity. Let $G$ be a topological group acting on a topological space $X$. The Serre spectral sequence associated to the Borel fibration $X\longrightarrow X_{G}\...
- 1,661
2 votes
0 answers
303 views
What is the filtration on the abutment of Künneth spectral sequence for higher algebraic K-theory?
I am learning the Künneth spectral sequence for higher algebraic K-theory of schemes from Roy Joshua's paper Algebraic K-theory and higher Chow groups of linear varieties. The setup is as follows. Let ...
- 721
4 votes
1 answer
221 views
Reference request: equivariant (co)homology
I am looking for an introductory book on equivariant (co)homology (with topological aspects). I want the following keywords to be included: reduced/unreduced Borel (co)homology, coBorel homology, Tate ...
- 427
4 votes
0 answers
287 views
Computing the cohomology algebra of a certain Hopf algebra
I would like to know the cohomology algebra $\text{Ext}_B^*(k,k)$ for a particular finite-dimensional Hopf algebra $B$ over a field $k$ of positive characteristic. It's easier to describe the dual $B^*...
- 4,429
4 votes
0 answers
283 views
Weaker version of Kudo's trangression theorem by Strom
In Chapter 33 of Strom's "Modern Classical Homotopy Theory", the author sketches the calculation of $H^*(K(\mathbb{Z}/p,*))$ following Postnikov's proof of the result. Crucially, this ...
6 votes
1 answer
274 views
Cohomology ring of the homotopy fiber of a self map of $S^2$ of degree 2
I am working on Exercise 67.7 of Lectures on Algebraic Topology by Prof. Haynes Miller, and I have got some difficulties. Let $f:S^2\to S^2$ be a map of degree 2, and let $F$ be its homotopy fiber. ...
- 155
3 votes
1 answer
321 views
Spectral sequence for cohomology of inverse limit of complexes
I have the following question. Let $\{C_i\}_{i\in I}$ be an inverse system of complexes with members in some abelian category $\mathcal{A}$ where all small limits exist + some technical conditions. ...
- 81
3 votes
0 answers
186 views
A question about the local coefficients system of the Borel fibration $X \rightarrow X_G \rightarrow B_G$
Let $G$ be a compact connected abelian group (not necessarily Lie group) and $X$ be a compact $G$-space. Consider the Borel fibration pairs $\left( X,X^{G}\right) \longrightarrow \left( X_{G},X_{G}^{G}...
- 1,661
3 votes
3 answers
391 views
Twisted cohomology from $\mathbb{Z}/2$ to $\mathbb{Z}/3$ with a nontrivial homomorphism
What is the twisted cohomology in the 3rd rank, $$H^3_{\rho}(B\mathbb{Z}/2,\mathbb{Z}/3)$$ where the $$\rho: \mathbb{Z}/2\to Aut(\mathbb{Z}/3)=\mathbb{Z}/2$$ is a nontrivial homomorphism?
- 10.8k
0 votes
0 answers
264 views
A question about the Leray spectral sequence
E. G. Sklyarenko give the next theorem in ''Some Applications of the theory of sheaves in general topology'' Sklyarenko The first line of page 47 states that '''Since this mapping has zero kernel for $...
- 1,661
3 votes
0 answers
187 views
For which fibrations the higher direct images of a constant etale sheaf constant as well?
Let $f:X\to Y$ be a (locally trivial) fibration of varieties over an algebraically closed field $k$. When are the corresponding E_2-terms for the corresponding Leray spectral sequence for the etale ...
- 17.5k
7 votes
1 answer
544 views
An issue with the construction of Steenrod powers
I'm trying to understand a construction of the Steenrod powers. It seems that there is a missing detail in the sources I'm reading. Let $p$ be a prime, $\pi$ denote the cyclic subgroup $\mathbb{Z}/p \...
- 183
3 votes
1 answer
221 views
An attempt at an alternative calculation of the rank of $\pi_n(MO)$
$\newcommand{\a}{\mathfrak a}\newcommand{\Z}{\mathbb Z}$Let $MO$ be the Thom Spectrum, then I am trying to come up with an alternative calculation that the rank of $\pi_n(MO)$ as a $\Z_2$ vector space ...
- 605
2 votes
0 answers
195 views
Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 1,136
0 votes
0 answers
126 views
Relation between Chow groups and K theory
I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence $$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
- 1,181
6 votes
0 answers
205 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
115 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 $r$...
3 votes
1 answer
291 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\...
- 1,912
3 votes
0 answers
119 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 ...
- 61
3 votes
1 answer
291 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 ...
1 vote
0 answers
176 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
131 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 (...
- 12.5k
2 votes
1 answer
287 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 ...
- 3,143
4 votes
1 answer
583 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 ...
- 1,661
4 votes
2 answers
410 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 $$\...
- 663
1 vote
0 answers
111 views
A question about the localization theorem of Borel-Hsiang and spectral sequence
Suppose that $T$ is a torus acting on a topological space $X $. Let $T\longrightarrow E_{T}\longrightarrow B_{T}$ be the universal $T$-bundle. Let $X\longrightarrow X_{T}\longrightarrow B_{T}$ be the ...
- 1,661
2 votes
2 answers
257 views
Extensions of $G$-modules parametrized by $H^1$
Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess one can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
- 339
13 votes
5 answers
2k views
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live ...
32 votes
1 answer
849 views
Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,312
2 votes
0 answers
200 views
Vanishing differential of Brown-Gersten-Quillen spectral sequence
Let $k$ be an algebraically closed field of characteristic zero and $X$ be an affine, simplicial toric 3-fold over the field $k$. I am trying to use the Brown-Gersten-Quillen spectral sequence to ...
- 721
10 votes
1 answer
339 views
Is there a correction to the failure of geometric morphisms to preserve internal homs?
Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
- 181
5 votes
1 answer
616 views
Two spectral sequences arising from a simplicial spectrum
Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization. Let's assume each $X_n$ is connective. From this situation, we can form two filtrations on $X$: the ...
- 401
2 votes
0 answers
240 views
Differentials in the Atiyah-Hirzebruch spectral sequence for a bounded generalized cohomology theory
$\newcommand{\res}{\mathrm{res}}$Let $X$ be a connected finite CW complex, and let $E$ be a bounded spectrum. For simplicity, let me assume that it has homotopy groups concentrated in degrees 0,1,2. ...
- 1,031
2 votes
0 answers
134 views
Alternative construction for the loop space (?)
There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...
- 2,192
1 vote
1 answer
727 views
Why should we study the total complex?
Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
- 165
2 votes
0 answers
214 views
Homotopy equivalence of chain complexes from subcomplexes and quotient complexes
Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
- 713
11 votes
2 answers
964 views
Spectral sequences and short exact sequences
Suppose I take a short exact sequence of filtered chain complexes: $$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$ We assume that $p$ and $q$ are filtration-preserving, so that $p(F_rA)\subseteq ...
- 769
1 vote
0 answers
137 views
Spectral sequence for a truncated semi cosimplicial space
Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
- 177
4 votes
1 answer
446 views
Collapse of spectral sequence computing Equivariant cohomology
I have already posted this question on math.stackexchanges but I got no answer and I decided to post it here. Let us consider the fibration $$ M\hookrightarrow EG\times_{\varphi}M\twoheadrightarrow BG ...
- 143
5 votes
0 answers
988 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}=\...
- 1,023
6 votes
1 answer
481 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 ...
- 2,192
3 votes
1 answer
481 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 ...
- 224