Questions tagged [spectral-sequences]
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411 questions
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Spectral sequence for double complexes
In this question they show the existence of a exact sequence involving page 2 terms. I want to know if the setup with a first quadrant double complex can be replaced by a double complex concentrated ...
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Wikipedia's version of Zeeman's comparison theorem for spectral sequences
Wikipedia states a version of Zeeman's comparison theorem for spectral sequences of flat modules over a commutative ring. For sources, one can look at the following.
Zeeman's original article
...
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Non-nilpotent elements in the cohomology of the mod-2 Steenrod algebra
I have heard that there are interesting non-nilpotent elements on the $E_2$-page of the mod-2 Adams spectral sequence (besides just $h_0$). For example, I have heard that $g$ and some related elements ...
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Hochschild-Serre spectral sequence with rational coefficients
Let $Z$ be the center of a group $G$. Does the second page of the Hochschild-Serre spectral sequence with rational coefficients look like :
$$E_2^{p,q}\simeq H^p(G/Z,\mathbb{Q})\otimes H^q(Z,\mathbb{Q}...
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Reference for Poincaré duality at the level of spectral sequences
Let $\xi = (E, B, F, p)$ be a (homologically simple) locally trivial fibre bundle. Assume that $E,F,B$ are all smooth oriented manifolds. (I am also interested in the case when at least $F$ is not ...
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Hochschild-Serre spectral sequence over the rationals
Does the Hoschild-Serre spectral sequence with rational coefficients give exactly the cohomology of the group, or the associated graded of the cohomology can be not isomorphic to the group cohomology ?...
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Local cohomology spectral sequence for quasi-coherent sheaves
Given an algebraic variety $X$ over a field $k$, a stratification
$$\emptyset = X_{-1} \subset X_0 \subset X_1 \subset \dotsb \subset X$$
so that $X_i \subset X_{i+1}$ is a closed immersion, and a ...
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Isomorphisms of sheaves of Abelian groups in Hesselholt's "Topological Hochschild Homology and the Hasse-Weil Zeta function"
This is from §5, p. 13, of Hesselholt's "Topological Hochschild Homology and the Hasse-Weil Zeta function".
The author claims there is a family of isomorphisms
\begin{equation*}
\phantom{\...
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Computing a certain group of homotopy classes of maps
Let, $G/PL \cong Y \times K(Z_2,6) \times K(Z,8) \times \cdots$. Where, $Y$ fits into the fibration sequence $K(Z,4) \to Y \to K(Z_2,2) \xrightarrow{\delta sq^2} K(Z,5) \to \cdots$ and $\delta$ is ...
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Introduction or Description of Mahowald's Sq1, Sq2 Diagrams
While reading Mahowald's paper on bo-Resolutions There are some very nicely illustrated modules on pg. 373:
I have seen this diagrams occasionally but I am curious if any one knows when and who (i.e. ...
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Eilenberg–Moore spectral sequence for dgas
Given $M$, $N$ dg-modules over a dg-algebra $A$ which is over a commutative unital ring $R$. There are spectral sequences:
$$E_{p,q}^2 = \operatorname{Tor}_{p,q}^{H^*(A)}(H^*(M),H^*(N)) \implies \...
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Two cohomology Cartan-Leray spectral sequences?
Let $X$ be a nice enough topological space and $p\colon X^{\prime}\rightarrow X$ a regular covering space with structure group $G$. Then, there is a Cartan-Leray homological spectral sequence $E^2_{p,...
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Spectral sequences every mathematician should know
Reading mathematical articles, I sometimes see how mathematicians pull out amazing spectral sequences seemingly at will. While many are built using standard techniques like exact couples or filtered ...
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Using spectral sequences to prove the Mayer Vietoris sequence on local cohomology is exact
In "On Some Local Cohomology Modules" by Lyubeznik, specifically Theorem 2.1, he gives a spectral sequence and writes "It is not hard to see that if n = 2, i.e. there are just two ...
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The Serre spectral sequence of the Borel fibration and monodromy action
Let $R$ be a commutative ring with unity. Let $G$ be a topological group acting on a topological space $X$. The Serre spectral sequence
associated to the Borel fibration $X\longrightarrow X_{G}\...
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