Questions tagged [spectral-sequences]
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388 questions
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Retrieval of algebra structure from spectral sequence
Suppose we have a spectral sequence of algebras and know that it degenerates at some $E_r$, take for example the cohomology Leray Serre spectral sequence associated to some fibration $F\hookrightarrow ...
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How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$
Hey everybody,
I think this question might be just a simple oversight on my part, but this has been bugging me a few days.
I am reading Hatcher's Spectral Sequences book, and trying to understand ...
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Fibrations with isomorphic Leray-Serre spectral sequences and non-isomorphic cohomology ?
Are there fibrations $F_i \to X_i \to B_i$ $(i=1,2)$ with path-connected bases $B_i$ and connected fibres $F_i$ such that their corresponding Leray-Serre spectral sequences (integral coefficients) are ...
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Construction of Serre Spectral Sequence
I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with ...
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Transgressions commute with the Steenrod operations on the base and fiber in a central group extension?
The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH:
In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence
...
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Spectral sequence for H-space bundles
Let $F \rightarrow E \rightarrow B$ be a fibre bundle such that $B$ is a smooth and compact manifold and $F$ obtains an associative H-space structure. Explicitly, it is not a principal bundle.
One ...
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Profiniteness Condition for Hochschild-Serre Spectral Sequence?
This question may seem elementary to experts but I am quite confused about it:
According to the entry of Lyndon–Hochschild–Serre spectral sequence on wikipedia, for a group extension $1\to N\to G\to ...
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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 $E_2$-...
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How to Compute Transgressions in a Serre Spectral Sequence?
For a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ there is an associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, which can be constructed by ...
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A Question on McCleary's book on Spectral Sequences
I am reading John McCleary's A User's Guide to Spectral Sequence and was quite confused about one result: On page 15 of the version I was reading, it says that if $E^{\star,\star}_2$ is the bigraded ...
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Computing the homology groups of spaces in a fibration
Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it ...
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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 ...
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On two spectral sequences for the cohomology of a double complex
For a (bounded) double complex (of abelian groups or vector spaces) one can consider two spectral sequences that converge to the cohomology of the totalization: one can first compute either the ...
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Generalized Beilinson spectral sequences
Assume we are workling on $\mathbb{P}^n$ for some $n\geq 1$ and we have a coherent sheaf $F$ on it.
Then there are two (well known?) spectral sequences $E_r^{p,q}$ with $E_1$-term:
$E_1^{p,q}=H^q(\...
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Multiplicativity in the descent spectral sequence
For a homotopy sheaf $\mathcal{F}$ of ring spectra over some space (/ site / whatever) $X$ with a cover $U_i$, we can build a "descent spectral sequence" with signature $$E^1_{p, q} = \pi_{p+q} \...
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On the multiplicative structure in spectral sequences.
Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
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Grothendieck spectral sequence [duplicate]
Possible Duplicate:
Composing left and right derived functors
Hi,
probably this question is obvious. I apologize for this.
Given functors $F$ and $G$ left exact, with as good properties as you ...
CommunityBot
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How does one view the De Rham spectral sequence as a Grothendieck spectral sequence?
I was rereading basic results on de Rham cohomology, and this led me inevitably to the fact that $H^q(X,\Omega^p)$ converges to $H^*(X)$ for any smooth proper variety (over any field). How does one ...
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How to compute the cohomology of the general linear group with integral entries
Q: So how does one compute the cohomology groups $H^*(GL_n(\mathbf{Z}),\mathbf{Z})$?
First note that $H^*(GL_n(\mathbf{Z}),\mathbf{Z})$ is isomorphic to $H_B^*(Y/GL_n(\mathbf{Z}),\mathbf{Z})$ (Betti ...
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The Hochschild-Serre spectral sequence relative to an ideal containing the derived subalgebra
Is the Hochschild-Serre spectral sequence $$H_\bullet(\mathfrak g/\mathfrak h,H_\bullet(\mathfrak h,k))\Rightarrow H_\bullet(\mathfrak g,k)$$ for an extension of Lie algebras $$0\to\mathfrak h\to\...
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Vanishing cohomology of line bundles on the Springer resolution
My question is regarding Broer's paper "Line bundles on the cotangent bundle of the flag variety" (see http://www.springerlink.com/content/t41418q436524515/). Given the Springer resolution, and its ...
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Special case of Leray spectral sequence
I am looking for a reference for what is stated in Srinivasan's book "Representations of Finite Chevalley Groups", which is apparently a special case of Leray spectral sequence. I'll quote the ...
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Third bordism group of BG, where G is an arbitrary compact Lie group.
Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ ...
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Slick Proof of Kudo Transgression Theorem
The Kudo Trangression Theorem has to do with the transgression in the Leray-Serre spectral sequence for cohomology in $\mathbb{Z}/p$ ($p$ odd). It can be proved by the method of the universal example,...
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Proof of the ''trangression theorem''
Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am ...
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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 ...
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Leray-Hirsch for HOMOLOGY?
Let $E\to B$ be a fibre bundle. The Leray-Hirsch theorem states under suitable assumptions, the cohomology of $E$ is an $H^*(B)$-module generated by suitable cohomology classes in $E$.
Is there any ...
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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 ...
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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 ...
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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} \...
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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 ...
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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 H_p(...
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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 ...
CommunityBot
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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} ...
CommunityBot
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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 ...
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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 ...
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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 $g:...
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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, ...
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