All Questions
Tagged with homological-algebra spectral-sequences
114 questions
2
votes
0
answers
137
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
4
votes
0
answers
86
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
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 ...
1
vote
0
answers
86
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 ...
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 ...
- 2,044
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 ...
- 339
27
votes
0
answers
468
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 $\...
- 1,021
5
votes
1
answer
471
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
1
vote
1
answer
510
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
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
856
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
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"
$$E_1^{i,j}=\...
- 711
1
vote
0
answers
86
views
Image of the boundary maps in the homological spectral sequence of a filtration of a chain complex
I'm trying to understand the construction of the homological spectral sequence of a filtration given in C.A.Weibel ''An introduction to homological algebra''. Here, they start with a filtration of a ...
- 911
0
votes
1
answer
238
views
Infinite dimensional homology and spectral sequences
I am new to spectral sequences, so I'm not sure about the difficulty of this question.
Suppose we have a filtration of a chain complex $\emptyset=D_{-1}\subset D_{0}\subset D_1\subset\dots D_n= C$ and ...
- 911
6
votes
1
answer
515
views
Leray spectral sequence and pullbacks
I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence:
Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E_2^...
- 177
1
vote
0
answers
167
views
Spectral sequence for two fibrations
Given maps of fibrations, i.e. commutative diagrams of smooth manifolds
$$\begin{matrix}
\ F & \to & E &\to & B \\\
\downarrow & & \downarrow & & \downarrow \\\
\ F'...
- 103
3
votes
0
answers
248
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
2
votes
0
answers
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
$$
E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...
- 519
1
vote
0
answers
93
views
Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence
Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
- 1,805
3
votes
0
answers
170
views
Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?
Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...
- 2,152
1
vote
0
answers
165
views
spectral sequence Ext(R/I,H^g(M)) => Ext^{p+q}(R/I,M)
I am reading papers of Local cohomology and came across some spectral sequences. I then started reading about spectral sequences from Rotman's book. I havent finished reading the chapters on spectrals ...
- 183
9
votes
1
answer
748
views
In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $F$ and $G$ on abelian categories:
\begin{align*}
E_{p,q}^{2}(A)=L_{p}G\circ L_{q}...
- 301
1
vote
0
answers
160
views
Diagrams filled by the edge homomorphism of the Grothendieck spectral sequence
Suppose $A,B,C,D,Z$ are abelian categories. Let $G:C\longrightarrow D$, $F:D\longrightarrow Z$, $P:C\longrightarrow A$, $G':A\longrightarrow B$, $P':D\longrightarrow B$ and $F':B\longrightarrow Z$ are ...
1
vote
0
answers
222
views
Cohomology spectral sequence of a CW complex filtered by its skeletons
Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$
is a filtration of $X$ by its skeletons $X^i$. Now ...
- 191
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'...
- 5,701
5
votes
1
answer
425
views
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
3
votes
0
answers
148
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
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
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
1
vote
0
answers
132
views
On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)
I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
- 642
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
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
7
votes
0
answers
538
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
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
2
votes
1
answer
259
views
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, ...
- 1,554
6
votes
3
answers
459
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
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
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 ...
1
vote
0
answers
174
views
Characterization of weakly convergence of spectral sequences
Let $C$ be a chain complex (in any abelian category) and let $\{F_p\}$ be a decreasing filtration of $C$. It induces a filtration on the homologies of $C$, given by $$F_pH=im(H(F_p)\rightarrow H(C)),$$...
- 183
6
votes
1
answer
371
views
Conditionally convergent spectral sequences with exiting and entering differentials
I have to deal with unbounded filtrations and want to use the conditional convergence of spectral sequences and the results from
[1]: J. Michael Boardman, Conditionally Convergent Spectral Sequences,...
- 466
5
votes
1
answer
185
views
Decompose $MT(E(d)\times_{\mathbb Z_2} SU(2))$ as the wedge sum or smash product of spectra
Consider the extension
$$1\to SU(2)\to X\to O\to1,$$
there are 4 possibilities for $X$:
$X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\...
- 2,147
3
votes
1
answer
299
views
Question about spectral sequences associated to filtered complexes with unbounded filtrations
All references below are from McCleary's book, second edition.
Suppose that we have a filtered complex where the filtration is unbounded. Suppose that the associated spectral sequence is weakly ...
- 205
4
votes
0
answers
396
views
Eilenberg-Moore spectral Sequence calculation
I want to use the cohomology Eilenberg-Moore spectral sequence to calculate the cohomology of the fibre of the map
$$
S^{n} \to \Omega S^{n+1}.
$$
Question 1: Is anyone aware of any references for ...
- 901
1
vote
0
answers
90
views
non zero differential in a spectral sequence
This is the situation:
Let $A = R_* \otimes C_*$ be an $R$-module where $C_*$ is a finitely generated graded ($*\geq 0$) vector space over a field $F$ which is also bounded above, and $R$ is a ...
- 81
4
votes
0
answers
207
views
Trivial action in the Hochschild-Serre spectral sequence
I probably don't understand something very basic about Hochschild-Serre spectral sequence.
Let $G$ be a group with normal subgroup $N$ and $M$ a $G$-module with trivial action. Then as far as I ...
4
votes
1
answer
332
views
Kernels and cokernels of multicomplex homomorphisms
Let $\mathcal A$ be a (complete and cocomplete) Abelian category.
A multicomplex in $\mathcal A$ is a bigraded object $X^{(\bullet,\bullet)}$ with differentials
$$
d^{(i,j)}_r\colon X^{(i,j)}\to X^{(...
- 2,452
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
votes
0
answers
174
views
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