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3
votes
2answers
271 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} ...
10
votes
2answers
831 views

Are the homology and cohomology Serre spectral sequences dual to each other?

If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...
1
vote
0answers
435 views

Spectral sequence for reduced homology

In the Serre spectral sequence, is it true that we can replace homology by reduced homology? That is: If $f:X\rightarrow B$ is a Serre fibration,with $F$ the fiber, then if $\tilde E^2_{pq}=\tilde ...
3
votes
1answer
413 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 ...
9
votes
2answers
726 views

Torsion in K-theory versus torsion in cohomology

Inspired by this question, I wonder if anyone can provide an example of a finite CW complex X for which the order of the torsion subgroup of $H^{even} (X; \mathbb{Z}) = \bigoplus_{k=0}^\infty H^{2k} ...
4
votes
1answer
481 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 ...
5
votes
1answer
802 views

Natural morphism appearing in Grothendieck spectral sequence

Assume we are in the setting of the Grothendieck spectral sequence (Weibel, 5.8): $G : A \to B, F : B \to C$ are left exact functors such that $G$ sends injective objects to $F$-acyclic objects. Now ...
3
votes
2answers
745 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, $E, F$ ...
5
votes
0answers
589 views

Multiplicative structure on spectral sequence

Let $E$ be a spectral sequence and assume that there is a product $E^{r}_{p_1,q_1} \times E^r_{p_2,q_2} \to E^r_{p_1+p_2,q_1+q_2}$ which satisfies the Leibniz rule (for all $p_i,q_i$, but $r$ ...
32
votes
5answers
5k views

Simple examples for the use of spectral sequences

I'm looking for basic examples that show the usefulness of spectral sequences even in the simplest case of spectral sequence of a filtered complex. All I know are certain "extreme cases", where the ...
25
votes
16answers
7k 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 ...
7
votes
0answers
967 views

An example computation of etale cohomology

(edited for clarity) In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...
6
votes
1answer
772 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
663 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 ...
12
votes
2answers
439 views

How do you compute the space of lifts of an E-infinity map?

Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps ...
5
votes
1answer
650 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, ...
41
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 ...
18
votes
5answers
2k views

Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?

One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...