# Questions tagged [spectral-sequences]

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### A question about spectral sequences

In the following proof (from The pontrjagin numbers of an orbit map and generalized G-signature theorem by Hsu-Tung Ku & Mei-Chin Ku https://link.springer.com/chapter/10.1007/BFb0085610), it is ...
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### Interpreting the edges in the Serre spectral sequence

Let $F \hookrightarrow E \stackrel{\pi}{\rightarrow} B$ be a fiber bundle. For simplicity, assume that $F$ is connected, that $B$ is $1$-connected, and that $B$ is a CW complex. Consider the Serre ...
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### Explicit description of the Leray spectral sequence with compact supports for a fibration

Consider a locally trivial fibration $f: E \to B$ with fiber $F = \mathbb{C}^n$. The Leray spectral sequence with compact supports is $$E_2: H^p_c(B, \underline{H^q_c(F)}) \implies H^{p+q}_c(E).$$ ...
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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 ...
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### Basic example of derived descent

I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example. Given a ...
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### Understand the proof that rational resolution is independent of the resolution

EDIT. Karl Schwede has given a different approach to solve this question, which is very clear. However, I still want to figure out how to deal with this problem along Ein's line, and waiting for the ...
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### Name for the "other term" in a derived exact couple

I'm building a spectral sequence using an exact couple $D^1 \to D^1 \to E^1 \to D^1$, with $k$th derived exact couple $D^k \to D^k \to E^k \to D^k$. In this case, I happen to have more information ...
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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 ...
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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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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 $E_1$? ...
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 \$\...