1
vote
0answers
153 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 ...
1
vote
0answers
157 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 ...
4
votes
1answer
132 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 ...
7
votes
2answers
332 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 ...
8
votes
2answers
532 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 ...
0
votes
2answers
308 views

An exercise about Tor [closed]

Let $I$ and $J$ be two ideals in $A$. Show that $\operatorname{Tor}_{1} (A/I, A/J) =\frac {I \cap J} { IJ} $ and $Tor_{2} (A/I, A/J) =\ker(I \otimes_ {A}J \to IJ )$. The first Tor is not a ...
3
votes
0answers
232 views

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 ...
0
votes
1answer
198 views

Spectral sequence for composition of global sections and tensor product of sheaves

Hi all, on the forum page http://www.groupsrv.com/science/about506648.html one can read the following (i cut out nonimportant parts): Question: Does anyone know any condition (non trivial) that ...
8
votes
3answers
768 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 ...
2
votes
0answers
272 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 ...
2
votes
1answer
124 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. ...
4
votes
1answer
442 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 ...
3
votes
1answer
372 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 ...
4
votes
2answers
482 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 = ...
2
votes
2answers
401 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 ...
3
votes
2answers
377 views

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 ...
4
votes
1answer
446 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 ...
5
votes
0answers
471 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 ...
2
votes
0answers
636 views

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 ...
77
votes
11answers
5k 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. ...
3
votes
2answers
267 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 ...
2
votes
1answer
132 views

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} ...
3
votes
1answer
378 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 ...
4
votes
1answer
459 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 ...
3
votes
2answers
700 views

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, $E, F$ ...
22
votes
16answers
6k 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 ...
3
votes
1answer
1k 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 ...
6
votes
1answer
667 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. ...
5
votes
1answer
629 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
votes
1answer
550 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, ...
38
votes
5answers
5k 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 ...