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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. ...
Vik78's user avatar
  • 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 $r$...
Joseph Sullivan's user avatar
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 ...
Elías Guisado Villalgordo's user avatar
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 ...
Elías Guisado Villalgordo's user avatar
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 ...
xir's user avatar
  • 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 ...
Conjecture's user avatar
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 $\...
FShrike's user avatar
  • 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 ...
Brian Shin's user avatar
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 ...
mrtaurho's user avatar
  • 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 ...
Faniel's user avatar
  • 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 ...
Richard Hepworth's user avatar
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}=\...
Gabriel's user avatar
  • 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 ...
Marcos's user avatar
  • 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 ...
Marcos's user avatar
  • 911
6 votes
1 answer
516 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^...
Victor de Vries's user avatar
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'...
UserIn's user avatar
  • 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 ...
Eric's user avatar
  • 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}...
user145752's user avatar
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 $\...
The Thin Whistler's user avatar
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 $\...
Andrea Marino's user avatar
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 ...
dongrugose's user avatar
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}...
Eric's user avatar
  • 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 ...
Pritthijit Biswas's user avatar
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 ...
Uncool's user avatar
  • 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'...
Pulcinella's user avatar
  • 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, ...
Faniel's user avatar
  • 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 ...
C. Zhihao's user avatar
  • 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 ...
Li Guanyu's user avatar
  • 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 ...
qqqqqqw's user avatar
  • 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 ...
Sergey Guminov's user avatar
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 ...
sdey's user avatar
  • 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\}_{...
Sumanta's user avatar
  • 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 ...
Sumanta's user avatar
  • 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 ...
Francesco Genovese's user avatar
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} \...
Federico Barbacovi's user avatar
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 ...
Jacob FG's user avatar
  • 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, ...
Pedro's user avatar
  • 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 = \...
Elise's user avatar
  • 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(...
Anh Dũng Lê's user avatar
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 ...
Onkar Singh Gujral's user avatar
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 $$F_pH=im(H(F_p)\rightarrow H(C)),$$...
Rafael's user avatar
  • 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,...
Pavel's user avatar
  • 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_{\...
annie marie cœur's user avatar
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 ...
Steve's user avatar
  • 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 ...
Niall Taggart's user avatar
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 ...
Vitolo's user avatar
  • 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 ...
David Levit-Gurevich's user avatar
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^{(...
Simone Virili's user avatar
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 ...
Sleipnir's user avatar
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}(\...
mrde05's user avatar
  • 39