All Questions
Tagged with spectral-sequences homological-algebra
114 questions
128
votes
12
answers
12k
views
Spectral sequences: opening the black box slowly with an example
My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials.
...
- 13.5k
56
votes
5
answers
9k
views
Why are spectral sequences so ubiquitous?
I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with ...
- 25.6k
44
votes
19
answers
16k
views
introductory book on spectral sequences
I have studied some basic homological algebra. But I can't send to get started on spectral sequences. I find Weibel and McCleary hard to understand.
Are there books or web resources that serve as ...
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
17
votes
2
answers
1k
views
Grothendieck spectral sequence when one of the functors is contravariant
Let $f \colon X \rightarrow S$ be a morphism of schemes. I am interested in computing the cohomology groups of
$$
\mathbf{R}\mathscr{H}om(\mathbf{R}f_* \mathcal{O}_X, \mathcal{O}_S)
$$
in terms of $\...
- 2,663
16
votes
1
answer
808
views
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...
- 863
15
votes
1
answer
2k
views
Has anyone seen this generalization of the snake lemma? Is it useful?
I originally posted this question on MSE (link), but was suggested to post here instead.
While learning about spectral sequences a friend of mine found a proof of the snake lemma using spectral ...
- 497
12
votes
4
answers
1k
views
Triply graded spectral sequence?
As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply,...
- 1,108
12
votes
3
answers
1k
views
Lyndon–Hochschild–Serre spectral sequence for a non-normal subgroup
Is there an analogue of the Lyndon–Hochschild–Serre spectral sequence for a non-normal subgroup?
What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$?
What is the best ...
- 385
12
votes
2
answers
523
views
A question on some computation of group cohomologies
Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...
- 123
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 ...
- 721
10
votes
3
answers
2k
views
Serre Spectral Sequence of Representations
Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...
- 4,920
10
votes
1
answer
657
views
Cap product on Leray-Serre spectral sequences
Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and $H^*(...
- 1,484
10
votes
0
answers
813
views
On functoriality of the Leray spectral sequence
The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
- 40.3k
9
votes
1
answer
748
views
In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $F$ and $G$ on abelian categories:
\begin{align*}
E_{p,q}^{2}(A)=L_{p}G\circ L_{q}...
- 301
9
votes
1
answer
455
views
Is this sequence of Lie algebra cohomology a part of spectral sequence?
There is an exact sequence
$$0 \to H^2(\mathfrak{g}, k) \to H^1(\mathfrak{g}, \mathfrak{g}^*) \to H^0(\mathfrak{g}, S^2\mathfrak{g}) \xrightarrow{d} H^3(\mathfrak{g}, k) \to H^2(\mathfrak{g}, \...
- 1,980
9
votes
2
answers
1k
views
H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory
Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...
- 10.5k
8
votes
1
answer
2k
views
How does one get the short exact sequence in a two-column spectral sequence?
In a two-column double complex, one gets from the associated spectral sequence short exact sequences $0\to E_2^{1,n-1}\to H^n\to E_2^{0,n}\to 0$, where $H^n$ is the cohomology of the total complex, ...
- 1,488
8
votes
1
answer
1k
views
Convergence of spectral sequences of cohomological type
Following the first chapter of Hatcher's great book "Spectral Sequences in Algebraic Topology", I got into problems with spectral sequences of cohomological type. Fix a ring $R$ once and for all. ...
- 727
8
votes
1
answer
615
views
Functorial construction of ("pre"-)spectral sequences? (Or - what is the "higher structure" underlying spectral sequences?)
Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the ...
- 7,799
8
votes
1
answer
363
views
Adams spectral sequence and short exact sequences. Some clarifications
as the title suggests I'm looking for some clarifications in the computations of the ext charts of some $A(1)$-modules arising as extensions of other modules. In particular, I've the following example ...
- 503
7
votes
2
answers
2k
views
isomorphic spectral sequences => quasi-isomorphic filtered chain complexes?
Let $(C,\partial)$ and $(C',\partial')$ be chain complexes of $R$-modules where $R$ is a (commutative) ring. Let $F$ and $F'$ be finite filtrations of $C$ and $C'$ respectively, i.e., $$\varnothing = ...
- 15.5k
7
votes
2
answers
4k
views
Conditions for the restriction $H^i(G,A)\to H^i(H,A)$ being surjective
I was wondering what the condition is for the restriction map (in group cohomology) $H^i(G,A)\to H^i(H,A)$ to be surjective.
I am a little confused about when maps between cohomology groups are ...
- 1,211
7
votes
1
answer
331
views
Does a filtered A_N algebra give rise to a multiplicative spectral sequence?
The question is pretty much in the title. It is a classical fact that a filtered dga gives rise to a multiplicative spectral sequence. It is claimed in Remark 4.1 of https://arxiv.org/pdf/1410.6728....
7
votes
0
answers
270
views
Differentials in spectral sequences and Massey products
Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...
- 965
7
votes
0
answers
541
views
Convergence of a spectral sequence of a double complex
In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
- 1,325
7
votes
0
answers
436
views
spectral sequence for a complex with two filtrations
Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
- 353
7
votes
0
answers
374
views
Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
- 7,377
6
votes
1
answer
858
views
What kind of spectral sequences come from double complexes?
Given a double complex in the first quadrant, one can derive from it a (homological or cohomological) spectral sequence converging to the (co)homology of the total complex of the double complex.
My ...
- 945
6
votes
3
answers
460
views
multiplicative structure of Ext
Basically, I am trying to compute something with the Adams spectral sequence (as a toy example). The $E^2$ page reduced to computing $Ext^{s,t}_{\Gamma} (\mathbb{F}_2, \mathbb{F}_2)$, where $\Gamma = \...
- 225
6
votes
1
answer
890
views
Serre spectral sequence with spectra
A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason other than ...
- 3,726
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^...
- 177
6
votes
1
answer
564
views
a question about Bockstein spectral sequence
I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459:
Question. for a fixed $k$, if $\beta$ does not hit $H_k(X;\...
- 1,990
6
votes
1
answer
371
views
Conditionally convergent spectral sequences with exiting and entering differentials
I have to deal with unbounded filtrations and want to use the conditional convergence of spectral sequences and the results from
[1]: J. Michael Boardman, Conditionally Convergent Spectral Sequences,...
- 466
6
votes
1
answer
458
views
Spectral sequence of a bicomplex equipped with a group action
Let $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ be complexes of $\mathbb{C}$-vector spaces.
We suppose that $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ are equipped with ...
- 101
6
votes
2
answers
503
views
Interpretations of differentials in hypercohomology spectral sequences as Yoneda products
I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups.
More ...
- 2,160
6
votes
1
answer
348
views
Weibel's H-book, Milnor's exact sequence for spectral sequence of filtered complex, Theorem 5.5.5
This is a question which I asked on StackExchange first, but might be more suited here.
I got stuck on the proof of Theorem 5.5.5 in Weibel's book. Not only that, but I also could not even find the ...
- 1,071
6
votes
1
answer
2k
views
cohomology version of Cartan-Leray spectral sequence that deduces cup product
On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:
If $X$ is a connected pace on which the group $\...
- 1,990
6
votes
1
answer
242
views
For which exact couples do associated spectral sequences degenerate at $E_1$?
It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My ...
- 16.9k
6
votes
1
answer
1k
views
Unbounded complexes, resolutions and computation of derived functors
Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded complex, I've heard that ...
- 456
6
votes
0
answers
237
views
A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?
This question is a follow-up to my previous question:
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
In that question, I discussed two different spectral sequences for ...
- 863
6
votes
0
answers
366
views
Transgression map spectral sequence of Ext
Let $X$ be a scheme over $k$ and $p:\ X \to Spec(k)$ the structure morphism. If $M$ is an étale sheaf of abelian groups over $Spec(k)$ I have a Grothendieck spectral sequence $$E^{p,q}_2=Ext^p_k(M,R^...
- 111
6
votes
0
answers
723
views
On the multiplicative structure in spectral sequences.
Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
- 21.8k
5
votes
1
answer
1k
views
Inverse limit of spectral sequences
I find myself in the following situation:
I have a sequence of first quadrant spectral sequences, let's call them $ E(n)_{p,q}^* $, each convergent to $E(n)_{p,q}^\infty$, with spectral sequence ...
- 651
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 ...
- 339
5
votes
1
answer
3k
views
Question about hypercohomology / spectral sequence of a complex of "almost-acyclic" sheaves
I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The ...
- 545
5
votes
1
answer
2k
views
Cohomology of derived tensor product of complexes and Künneth spectral sequence
Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive ...
- 2,079
5
votes
1
answer
186
views
Decompose $MT(E(d)\times_{\mathbb Z_2} SU(2))$ as the wedge sum or smash product of spectra
Consider the extension
$$1\to SU(2)\to X\to O\to1,$$
there are 4 possibilities for $X$:
$X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\...
- 2,147
5
votes
1
answer
918
views
colimits of spectral sequences
I'm looking for some references about colimits of spectral sequences.
More precisely: let $X : I \longrightarrow \cal{C}$ be a functor from a filtered category $I$ to the category of double cochain ...
- 1,975
5
votes
1
answer
609
views
Functoriality of filtered spectral sequences
What is the appropriate functoriality statement of a filtered chain map between filtered spectral sequences?
Suppose that we have two filtered chain complexes $C,C'$ and a filtered chain map $f\colon ...
