All Questions
Tagged with spectral-sequences homological-algebra
114 questions
5
votes
1
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425
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Can we construct a filtered chain complex from a spectral sequence?
Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, ...
- 673
5
votes
2
answers
651
views
Inflate a finite-group cocycle into coboundary in non-Abelian groups
Edit: In case that there is no solution for the original question, I modify to enrich the question.
We like to ask a possible specific inflation a $H^3(Q, \mathbb{R} /\mathbb{Z})$ cocycle with a ...
- 755
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"
$$E_1^{i,j}=\...
- 773
5
votes
0
answers
170
views
multiplication in spectral sequence
I am trying to understand this paper. Let $M$ be a compact Kaehler manifold of dimension $n$, $X$ is a holomorphic vector field, $i_X$ the contraction operator, i.e. for $\alpha$ a $p$-form, then $i_X(...
- 315
5
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0
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544
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Strong Convergence vs Conditional Convergence for Spectral Sequences (Is there a simple explanation?)
I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?
(Hopefully a simpler explanation than ...
- 1,229
5
votes
0
answers
521
views
Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
- 16.9k
5
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0
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675
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Do exact functors commute with spectral sequences ?
Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let
$$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corresponding spectral ...
- 16.2k
4
votes
1
answer
2k
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Tensor product of spectral sequences?
I'm wondering about a cross product for spectral sequences. I've got an idea, and I wonder if it is written up anywhere, or if it even holds water.
Let's start with three spectral sequences, $E, F$ ...
- 12.5k
4
votes
2
answers
1k
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Tracking spectral sequence differentials
I read a number of posts here on MO, but haven't quite found an answer to the question of where the differentials in a spectral sequence come from.
I came across a differential $d^{0,1}$ on the $E_2$-...
- 1,211
4
votes
1
answer
332
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Kernels and cokernels of multicomplex homomorphisms
Let $\mathcal A$ be a (complete and cocomplete) Abelian category.
A multicomplex in $\mathcal A$ is a bigraded object $X^{(\bullet,\bullet)}$ with differentials
$$
d^{(i,j)}_r\colon X^{(i,j)}\to X^{(...
- 2,452
4
votes
1
answer
195
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Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?
I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H_*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$.
How about for multiple summands in the ...
- 1,229
4
votes
0
answers
88
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
0
answers
397
views
Eilenberg-Moore spectral Sequence calculation
I want to use the cohomology Eilenberg-Moore spectral sequence to calculate the cohomology of the fibre of the map
$$
S^{n} \to \Omega S^{n+1}.
$$
Question 1: Is anyone aware of any references for ...
- 901
4
votes
0
answers
207
views
Trivial action in the Hochschild-Serre spectral sequence
I probably don't understand something very basic about Hochschild-Serre spectral sequence.
Let $G$ be a group with normal subgroup $N$ and $M$ a $G$-module with trivial action. Then as far as I ...
4
votes
0
answers
420
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Cohomology of double complex with exact rows
Let $(C^{p,q},d_h,d_v)$ be an unbounded double complex of modules of some algebra $A$ in an abelien category. Let $d_h,d_v$ be the horizontal and vertical differential respectively. Suppose that
(1) ...
- 205
4
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0
answers
1k
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Grothendieck spectral sequence [duplicate]
Possible Duplicate:
Composing left and right derived functors
Hi,
probably this question is obvious. I apologize for this.
Given functors $F$ and $G$ left exact, with as good properties as you ...
- 51
3
votes
1
answer
299
views
Question about spectral sequences associated to filtered complexes with unbounded filtrations
All references below are from McCleary's book, second edition.
Suppose that we have a filtered complex where the filtration is unbounded. Suppose that the associated spectral sequence is weakly ...
- 205
3
votes
1
answer
678
views
Convergence of right half-plane spectral sequence bounded on the right
This is a sequel to my previous question colimits of spectral sequences .
I think I've found the answer in S.A. Mitchell's paper "Hypercohomology spectra and Thomason's descent theorem". There the ...
- 1,975
3
votes
1
answer
260
views
Strong convergence of whole-plane spectral sequences
I am trying to understand strong convergence for whole-plane spectral sequences in the paper by J.Boardman:
https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-...
- 205
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 ...
3
votes
1
answer
465
views
What bigrading is used in this spectral sequence?
I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it ...
- 1,564
3
votes
2
answers
493
views
Is the first filtration Hausdorff?
Maybe this is too technical and elementary, but I cannot make up my mind, nor find a reference.
The situation is the following: let $X$ be a double cochain (right half-plane) complex of abelian ...
- 1,975
3
votes
0
answers
249
views
Explicit computation of hyper Ext in terms of the homologies of the input chain complexes
This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
- 301
3
votes
0
answers
170
views
Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?
Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...
- 2,152
3
votes
0
answers
277
views
Dimension three spectral sequences
If you have two triangulated functors $F,G$, you can compute the cohomology of $FG(-)$ in terms of the cohomology of $G(-)$ and the cohomology of $F(-)$, in terms of a ``Grothendieck spectral sequence'...
- 5,711
3
votes
0
answers
149
views
Group cohomology with coefficients in a graded module
I am working in a problem group cohomology and nailed it down to compute a cohomology using an spectral sequence argument. The situation is as follows:
Let $G = C_4 = \langle \sigma \rangle$ be the ...
- 283
3
votes
0
answers
101
views
Geometric filtration for Eilenberg-Moore spectral sequence
I'm reading the paper by Eilenberg-Moore (https://link.springer.com/content/pdf/10.1007/BF02564371.pdf) about the Eilenberg-Moore spectral sequence.
In section 11, they introduce the notion of ...
- 449
3
votes
0
answers
71
views
Deformations of nilpotent parts of superalgebras
I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598
After ...
- 95
3
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0
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174
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Induced Homomorphism on Cohomology of Symmetric Group 3
For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...
- 39
3
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0
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166
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Edge map in derived categories
Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...
- 5,670
3
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0
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310
views
Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
- 21.8k
2
votes
2
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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 ...
- 339
2
votes
1
answer
275
views
Why only consider decreasing filtrations on cochain complexes?
When reading various literature on spectral sequences one always comes across two setups:
A chain complex with an increasing filtration
A cochain complex with a decreasing filtration
My question is ...
- 2,788
2
votes
1
answer
186
views
Can I bound the degree of a contracting homotopy in an exact filtered complex?
Suppose that I have given you a bigraded vector space $V = \bigoplus_{i,j} V_{i,j}$. The first grading is a "homological" $\mathbb Z$-grading, and the second is an independent $\mathbb Z$-grading. ...
- 54.6k
2
votes
1
answer
264
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Trivialize a cup-product 3-cocycle of $G$ in a larger group $J$
Inspired by this question, let us take a nontrivial 3-cocycle $\omega_3^G(g_a, g_b, g_c) \in H^3(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. ...
- 755
2
votes
1
answer
259
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Poset filtrations
Consider a chain complex $C$ and a poset $P$ so that there is a filtration by subcomplexes $C^p$ of $C$ where $p\in P$ in such a way that $p<q$ implies $C^p \leqslant C^q$.
As a second option, ...
- 1,564
2
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1
answer
247
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Colimit of intersections
Let $B_i^p$ be a family of sets, where $p\in \mathbb{N}$ and $i \in I$, $I$ being a directed set, and such that, for every $i$, we have a descending chain of inclusions
$$
\dots \supset B_i^{p-1} \...
- 1,975
2
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0
answers
138
views
Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 658
2
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0
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147
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Homotopy equivalence of chain complexes from subcomplexes and quotient complexes
Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
- 673
2
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0
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193
views
When does Tate spectral sequence degenerate at $E_2$?
For a spectrum $M \in \text{Sp}^{B\mathbb{S}^1}$ with a circle group action, there is Tate spectral sequence
$$
E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...
- 519
2
votes
0
answers
269
views
Dress' construction and Serre spectral sequence
Currently, I am reading Serre spectral sequence, given below, using Dress' construction.
Let $f:E\to B$ be a Serre fibration. Then, there is a first quadrant
spectral sequence $\big\{E^r,d^r\}_{...
- 632
2
votes
0
answers
486
views
An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology
I am currently reading Künneth spectral sequence, which is given below.
Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...
- 632
2
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0
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108
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Relating inflation maps from spectral sequences in lower and higher dimensions
The spectral sequence has some nice property.
Consider $ N \to G \overset{R}{\to} Q $ and $G/N=Q$. There is a spectral sequence $\{E^{p,q}_n, d_n\}$ with: (i) The differential is defined as a map $...
- 10.5k
2
votes
0
answers
216
views
completion and convergence of spectral sequence
I would like to understand the connection between $p$-adic completion and the strong convergence of a spectral sequence. Precisely, suppose $E^2_{s,t}\implies G_{s+t}$ is a first quadrant strongly ...
- 21
2
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0
answers
1k
views
What is a Beilinson spectral sequence?
I'm writing to ask just a question. I would like to understand better what is the Beilinson's spectral sequence and how it can be used. Is there any useful and clear reference you advice to someone ...
- 21
2
votes
0
answers
747
views
What is the abutment filtration of the second spectral sequence of hypercohomology?
I have been recently learning about spectral sequences, following mainly Illusie's notes and EGA, and I am about to write some expository notes, but there are still some points that I was not able to ...
- 142
2
votes
0
answers
478
views
Morphisms of Spectral Sequences and alternating products
Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences.
Assume that morphisms $...
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 ...
- 165
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
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