All Questions
Tagged with spectral-sequences reference-request
36 questions
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80
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Relation between Chow groups and K theory
I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence
$$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
- 613
2
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1
answer
191
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Cohomology class of fiber bundle
Suppose I have a fiber bundle $\pi: E\rightarrow B$, with fiber $F$, such that the Serre spectral sequence on cohomology is immediately degenerate. In other words, $H^*(E)=H^*(B)\otimes H^*(F)$.
I ...
- 3,093
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47
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Generalized edge map in spectral sequence of double complex
suppose we have a cohomologically indexed double complex $C^{\bullet,\bullet}$ with its spectral sequence
$$E_2^{p,q}=H^p_{vert}(H^q_{horiz}(C))\Rightarrow H^{p+q}(C)$$
and suppose that the horizontal ...
- 2,054
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107
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Spectral sequence for a truncated semi cosimplicial space
Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
- 177
4
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2
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409
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Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
- 441
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1
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518
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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^...
- 177
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249
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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 ...
- 301
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1
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233
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Mayer–Vietoris sequence for coproduct of Hopf algebras
Is there a Mayer–Vietoris-type sequence for the homology of a coproduct of two Hopf algebras over an ideal? The definition of the coproduct can be found in Agore - Categorical Constructions for Hopf ...
- 1,374
2
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164
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Exact sequence for low-degree terms of relative de Rham cohomology
Let $\varphi:X\to Y$ be a morphism of schemes, smooth of relative dimension 1. The de Rham cohomology $H^\bullet(X/Y)$ is the result of applying the derived functor $R^\bullet\varphi_*$ to the de Rham ...
- 1,338
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174
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When the Leray spectral sequences for nice compactifications give the Deligne's weight ones?
Assume that $X$ is a proper smooth variety over an algebraically closed field $k$, $U=X\setminus (\cup D_i))$ where $D_i$ are closed subvarieties such that the set-theoretic intersections of all sets ...
- 16.9k
3
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277
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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'...
- 5,711
5
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1
answer
425
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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, ...
- 673
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3
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914
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Spectral sequences in algebraic topology [duplicate]
What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a ...
4
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1
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315
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Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request
I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...
- 301
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132
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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 ...
- 642
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1
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2k
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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 ...
- 2,079
14
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404
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Computations using "Stover's spectral sequence"
In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces.
The second ...
- 208
5
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1
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609
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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 ...
8
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267
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A diagram in the proof of Theorem 2.5.5 of 'Cohomology of Number Fields' and the Tate Spectral Sequence
I've been reading the book 'Cohomology of Number Fields' for years.
But I couldn't check the commutativity of the diagram
on page 126 until now. So I ask for help.
The diagram is induced by taking ...
- 1,013
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448
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Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
- 1,713
3
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0
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166
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Edge map in derived categories
Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...
- 5,670
3
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165
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Reference for specific detail on Serre spectral sequence
In "A primer on spectral sequences" http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf (by J.P.May apparently, although no name is given in the pdf) I found a very detailed version of the ...
- 404
4
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1
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227
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How are p-primary parts determined for odd p?
When looking at surveys of computations of the homotopy groups of spheres there is a common theme. All the odd primary parts are thrown away.
How are odd primary part calculations done in relation ...
- 41
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1
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441
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Reference request: cohomology of Eilenberg Maclane spaces with $p$-local groups
In Rudyak's 'On Thom Spectra, Orientability, and Cobordism', the following fact is used:
Let $\mathbb{Z}_{(p)}$ be $\mathbb{Z}$ localized at the ideal $(p)$. Let $\pi,\tau$ be two cyclic $\mathbb{...
- 235
6
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1
answer
241
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For which exact couples do associated spectral sequences degenerate at $E_1$?
It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My ...
- 16.9k
5
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0
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521
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Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
- 16.9k
3
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310
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Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
- 21.8k
3
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241
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Eilenberg-Moore spectral sequence for path-loop fibration over Q\Sigma X (reference request)
Related to the question here, here is another question. Consider the kernel of the map $H_*(QY;Z/p)\rightarrow H_{*+1}(Q\Sigma Y;Z/p)$. restricted to
$PH_*(QY)$, and let's say $Y$ itself is a ...
- 3,051
10
votes
1
answer
657
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Cap product on Leray-Serre spectral sequences
Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and $H^*(...
- 1,484
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1
answer
1k
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Multiplicative Structure of the Atiyah-Hirzebruch/Leray-Serre spectral sequence
This is related to this question (edit: now answered, see below). Is there a nice explanation of the multiplicative structure on the higher pages of that spectral sequence? I want to assume that $h$ ...
- 6,813
12
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2
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523
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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 ...
- 123
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675
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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 ...
- 16.2k
8
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1
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739
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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
...
- 83
5
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1
answer
2k
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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(\...
- 1,391
6
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0
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723
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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 ...
- 21.8k
6
votes
0
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418
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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\...
- 47.3k