Questions tagged [spectral-sequences]
The spectral-sequences tag has no usage guidance.
388 questions
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Grothendieck spectral sequence and exact couples
I'm thinking about the Grothendieck spectral sequence. I'm trying to understand how it relates to exact couples. Can anyone help me prove in general that the Grothendieck Spectral sequence converges ...
user30211
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When does Tate spectral sequence degenerate at $E_2$?
For a spectrum $M \in \text{Sp}^{B\mathbb{S}^1}$ with a circle group action, there is Tate spectral sequence
$$
E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...
- 519
4
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1
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Higher order differentials of Bockstein spectral sequence
The Bockstein SS is obtained from the exact sequence
$$0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$$
with $E_1^p=H^p(X,\mathbb{Z}/2)$ and the differential $d_1=Sq^1$.
How to identify ...
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3
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Tensor product of exact couples
Suppose $(D_1,E_1;i_1,j_1,k_1)$ and $(D_2,E_2;i_2,j_2,k_2)$ are two exact couples, ie., there are exact sequences $D_1\xrightarrow{i_1}D_1\xrightarrow{j_1}E_1\xrightarrow{k_1}D_1$ and $D_2\xrightarrow{...
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The $\pi_1(BM)$ action on $H^*(BH,R)$ in a Serre fibration $BH\to BG\to BM$
$R$ is a ring. Applying the Serre spectral sequence to a fibration $F\to E\to B$, to avoid local coefficients, we need to require that $\pi_1(B)$ acts on $H^*(F, R)$ trivially. Consider a fibration of ...
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Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence
Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
- 1,805
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1
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$BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$
What is know about the homotopy groups of $S/3$ where $S/3 = \mathrm{hocofib}(S \xrightarrow{\cdot 3} S)$? Otherwise, is there some reference I can consult for the $BP$-ANSS for $S/3$?
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The Hochschild–Serre spectral sequence and cup products
Let $X$ be a variety over a field $k$ with separable closure $k_s$. Let $A$, $B$ be étale sheaves on $X$. Consider now the Hochschild–Serre spectral sequences.
\begin{align*}
E_2^{pq}: H^p(k, H^q(X_{...
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Frölicher spectral sequence of a surface
Asked this on MSE but didn't get much attention.
Let $ S $ be a compact complex surface. Can anyone provide a proof of the fact that the Frölicher spectral sequence of $ S $ degenerates at $ E_1 $?
...
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Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?
Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...
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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
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spectral sequence Ext(R/I,H^g(M)) => Ext^{p+q}(R/I,M)
I am reading papers of Local cohomology and came across some spectral sequences. I then started reading about spectral sequences from Rotman's book. I havent finished reading the chapters on spectrals ...
- 183
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In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $F$ and $G$ on abelian categories:
\begin{align*}
E_{p,q}^{2}(A)=L_{p}G\circ L_{q}...
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How to compute the $G$-theory groups of $k[x,y]/(xy)$ for any field $k$
I am trying to compute the $G$-theory groups of the ring $k[x,y]/(xy)$ for any field $k$. What I have tried so far are two approaches.
Approach 1: Use the $G$-theory localization sequence for $k[x,y]/(...
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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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Does Kudo's transgression theorem still hold when the coefficient is ℤ/pⁿℤ?
I am trying to compute a Lyndon–Hochschild–Serre spectral sequence with coefficient $\mathbb{Z}/p^2\mathbb{Z}$, where $p$ is an odd prime, and Kudo's transgression theorem looks like a good tool for ...
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Borel spectral sequence with non-compact fibers
I was reading the following paper which claims to generalize Borel spectral sequence for non-compact Stein fibers. However, I don't understand how the following bundle fits into the picture:
$$
(\...
- 1,374
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160
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Diagrams filled by the edge homomorphism of the Grothendieck spectral sequence
Suppose $A,B,C,D,Z$ are abelian categories. Let $G:C\longrightarrow D$, $F:D\longrightarrow Z$, $P:C\longrightarrow A$, $G':A\longrightarrow B$, $P':D\longrightarrow B$ and $F':B\longrightarrow Z$ are ...
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Cohomology ring $H^*(BG,\mathbb{Z}_2)$ for $G=\mathbb{Z}_2\ltimes B^2\mathbb{Z}^2$
$$
\newcommand{\Z}{\mathbb{Z}}
$$
Consider the group $G=\Z_2\ltimes B^2\Z^2$ where $\Z_2$ acts on $\Z^2$ by interchanging the two factors of $\Z^2$, hence $\Z_2$ also acts on the Eilenberg-MacLane ...
- 10.5k
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Borel's transgression theorem for spectral sequences
My question is about part of the Borel Transgression Theorem on spectral sequences, translated from [1, Theorem 13.1] (see also [2] for a translated version of the whole paper):
Let $B^\bullet := \...
user126256
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0
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222
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Cohomology spectral sequence of a CW complex filtered by its skeletons
Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$
is a filtration of $X$ by its skeletons $X^i$. Now ...
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Thom spectrum of $(\mathrm{Spin}\times_{Z_2} \mathrm{SO}(d))$
$\DeclareMathOperator\MSO{MSO}\DeclareMathOperator\MSpin{MSpin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}\DeclareMathOperator\BG{BG}\DeclareMathOperator\BO{BO}\DeclareMathOperator\MG{MG}\...
- 487
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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
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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'...
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Galois-action on spectral sequence
Let $X_\bullet\to S$ be a proper surjective hypercover of a $k$-scheme by smooth proper $k$-schemes. This gives a proper surjective hypercover $X'_\bullet\to S_{\bar{k}}$ where $X'_n:=X_n\times_k \bar{...
- 4,418
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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, ...
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Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity
Suppose $G$ is a Lie group, with $\pi_0(G)$ not necessarily finite, but might as well assume $G_0$, the connected component of the identity, is compact.
In the case that $\pi_0(G)$ is finite, then we ...
- 35.5k
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714
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Spectral sequence from a stratification by closed subvarieties
I am looking for a reference for the following result: If $X$ is an algebraic variety and
$$X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$$
is a stratification (edit: filtration) ...
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Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$
First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
- 101
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Group cohomology with coefficients in a graded module
I am working in a problem group cohomology and nailed it down to compute a cohomology using an spectral sequence argument. The situation is as follows:
Let $G = C_4 = \langle \sigma \rangle$ be the ...
- 283
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101
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Geometric filtration for Eilenberg-Moore spectral sequence
I'm reading the paper by Eilenberg-Moore (https://link.springer.com/content/pdf/10.1007/BF02564371.pdf) about the Eilenberg-Moore spectral sequence.
In section 11, they introduce the notion of ...
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Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 64k
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votes
3
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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 ...
8
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2
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492
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Conditions under which the preimage of a submanifold in nontrivial in homology
Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-...
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About an argument in Olsson's book
The following picture is from Algebraic spaces and stacks (p.54) by Martin Olsson.
I don't understand how to conclude that $\alpha$ is induced by a nonzero class in the end.
It seems that there might ...
- 1,906
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209
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Cohomology of doubly pinched torus via spectral sequences
Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
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Differentials in spectral sequences and Massey products
Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...
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$\pi_{2p^kn - 1}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}_{p^{r+1}}$ summand
I am reading Neisendorfer's paper Samelson products and exponents of homotopy groups and related papers. I am stuck on theorem 14.1 on page 21, which says that there exists a $\mathbb{Z}_{p^{r+1}}$ ...
- 1,263
3
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1
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Prove or disprove that there exists no $G$ structure with its bordism group $\Omega_1^{G} =\mathbb{Z}/N$ for $N>2$
It can be found that there are the following bordism group $\Omega_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds:
$$
\Omega_0^{SO} = \mathbb{Z} , \quad \Omega_1^{SO} = ...
- 2,147
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348
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Weibel's H-book, Milnor's exact sequence for spectral sequence of filtered complex, Theorem 5.5.5
This is a question which I asked on StackExchange first, but might be more suited here.
I got stuck on the proof of Theorem 5.5.5 in Weibel's book. Not only that, but I also could not even find the ...
- 1,071
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1
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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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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 ...
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1
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529
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Weak Lefschetz theorem for Lef line bundles
I'm studying
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
The premises are the following....
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201
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Using the Serre spectral sequence - moving between $\mathbb{Z}/2$ and $\mathbb{Z}$ information
I am trying to understand the computation of $\pi_5(S^3)$ and $\pi_6(S^3)$ using the Serre spectral sequence. I know already that $\pi_5(S^3)$ is only 2-torsion and $\pi_6(S^3)$ is 2-torsion together ...
- 5,349
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269
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Dress' construction and Serre spectral sequence
Currently, I am reading Serre spectral sequence, given below, using Dress' construction.
Let $f:E\to B$ be a Serre fibration. Then, there is a first quadrant
spectral sequence $\big\{E^r,d^r\}_{...
- 632
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0
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486
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An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology
I am currently reading Künneth spectral sequence, which is given below.
Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...
- 632
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1
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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
3
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0
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186
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Cobordism theory of some weird space
Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$.
The $W$ is a homogeneous space (also a quotient space), but not a group.
Previously, I am aware of the ...
- 10.5k
2
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0
answers
163
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A Thom isomorphism for sheaves
Let say $\mathcal{F}$ is a locally free sheaf of abelian groups over $X$, where $X$ is an algebraic variety over $\mathbb{C}$ (or a field $k$) with analytic (or étale) topology and $Z$ is a closed ...
7
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541
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Convergence of a spectral sequence of a double complex
In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
- 1,325