Spectral sequences compute homology groups by taking a sequence of approximations, and generalise exact sequences. They find application in algebraic topology.
Questions tagged [spectral-sequences]
434 questions
9
votes
1 answer
Easy spectral sequence
This is a question in Weibel's Homological Algebra.
Suppose that a spectral sequence converging to $H_\ast$ has $E^2_{pq} = 0$ unless $p = 0,1$. Show that there are exact sequences
$$0 \rightarrow E^2_{1,n-1} \rightarrow H_n \rightarrow E^2_{0,n}…
sumo
- 91
5
votes
1 answer
Spectral sequences from Cartan-Eilenberg systems
This is an exercise from Mcleary's book on Spectral sequence which I have been stuck with for some time.
Let us recall what a Cartan-Eilenberg system is:
IT consists of a module $H(p,q)$ for each pair of integers, $-\infty \leq p \leq q \leq \infty$…
Tedar
- 529
3
votes
1 answer
Collapse of spectral sequence computing Equivariant cohomology
Let us consider the fibration
$$
M\hookrightarrow EG\times_{\varphi}M\twoheadrightarrow BG
$$
where $M$ is a $G-$space, $\varphi$ is the action of $G$ on $M$
and $EG\times_{\varphi}M$ is the diagonal quotient. It is required
that $M$ is compact and…
RiemannGauss
- 199
2
votes
1 answer
Spectral Sequences - John McCleary Example 1.A (First Quadrant Topological Spectral Sequence)
On John McCleary's book "A user's guide to sepctral sequences", on page 6 he gives the following example:
Example 1.A. Suppose that there is a first quadrant spectral sequence of cohomologica type with initial term $(E_2^{*,*},d_2)$, converging to…
D18938394
- 1,210
2
votes
1 answer
Elementary (?) question on differentials in a spectral sequence
Suppose I have a chain complex $C$ with differential $D$ and filtration $F$. Suppose further that I can decompose $D$ by its action on the filtration, i.e. there are maps $D = D_1 + D_2 + \cdots$ so that $D_k: F^iC \to F^{i+k}C$. (In every example…
Adam Saltz
- 2,596
0
votes
0 answers
Why $\ker \, d = k^{-1} (\ker \,j)$?
Here is the part of the book (user's guide to spectral sequences) I am reading:
I have the following questions about this part:
1- Why we are sure that $k$ has an inverse (the user used $k^{-1}$)?
2- if $d = j \circ k$ why $\ker d = k^{-1}(\ker…
Intuition
- 3,269
0
votes
1 answer
An exercise of spectral sequences in Vakil's notes.
This is an exercise in 1.7. Spectral sequences of his Foundation of Algebraic Geometry
where $E^{\bullet,\bullet}$ is a double complex with $\mathrm{d}_{\to}:E^{p,q}\to E^{p+1,q}$ and $\mathrm{d}_\uparrow:E^{p,q}\to E^{p,q+1}$. The single complex…
Display Name
- 1,373