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Let $F \to E \to B$ be a fibration with $B$ simply connected (more generally, such that $\pi_1(B)$ acts trivially on the homology of $F$). Then there is a Serre spectral sequence $H_p(B, H_q(F)) \to H_{p+q}(E)$. One can do the same for singular cohomology. However, for reasonable spaces (specifically, locally contractible spaces, e.g. CW complexes), singular cohomology is the same as sheaf cohomology of the constant sheaf $\mathbb{Z}$.

But there is another spectral sequence for sheaf cohomology: the Leray spectral sequence. Given spaces $X, Y$ and $f: X \to Y$, and a sheaf $\mathcal{F}$ on $X$, there is a spectral sequence $H^p(Y, R^q_f(\mathcal{F})) \to H^{p+q}(X, \mathcal{F})$. The Wikipedia article hints that the topological implications of this include in particular the Serre spectral sequence. I would be interested in this, because I like the machinery of the Grothendieck spectral sequence (from which the Leray spectral sequence easily follows), and would be curious if the Serre spectral sequence could be obtained as a corollary.

Is this possible?

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    What happens if you try to check the hypotheses of the Grothendieck s.s. theorem in the case of the Serre s.s. for cohomology?2010-11-21
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    (By the way, note that even if using the Grothendieck s.s. you manage to get a s.s. whith $E_2$ term of the same shape as that of the Serre s.s., that is not enough (sadly!) to know that the s.s. you got is the same one as the one Serre got.)2010-11-21
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    @Akhil, Serre did precisely as you suspect. He looked at the Leray spectral sequence and applied it to a fibration over a CW complex, and noticed that it cleans up quite a bit in that situation. If you read Serre's papers or Dieudonne's history you'll see this.2010-11-21
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    Tohoku is from 57, Serre's thesis from 51, and Leray's work predates that. The original arguments, Akhil, did not involve Grothendieck's s.s.2010-11-21
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    @Ryan: Sure, I'll take a look at _Homologie des espaces fibres_ and see how it goes; thanks for the recommendation (the idea hadn't occurred to me).2010-11-21
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    @Mariano: Sure, but I just like to think of the Leray SS as a special case of the Grothendieck SS, even if Leray's came before Grothendieck's.2010-11-21
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    *What* are you asking exactly? If Serre's ss is a corollary of Leray's, or if (both?) follow from Grothendieck's.2010-11-21
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    The former. Isn't Leray's a direct corollary of Grothendieck's? (Take the first functor to be the pushforward $f_*$ from $Sh(X) \to Sh(Y)$ and the second to be global sections. $f_*$ preserves flabbiness, so Grothendieck ss hypotheses apply.)2010-11-21
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    Akhil, as I said before, it follows from G's s.s. that a spectral sequence with the same initial term as Leray's exists. But a s.s. is much more than its initial term!2010-11-21
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    @Mariano: Good point. All I have ever seen used is the initial term and the existence of a product structure for the cohomology SS (which may be derivable in the Grothendieck case from cup-products). I have so far been able to avoid thinking about the differentials (except in deducing what they are from $E_\infty$, e.g. if the fiber space is contractible), but I have not gotten very far in this subject.2010-11-21
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    I really would like to see an answer here which does not just say "it's obvious". After all, the Leray spectral sequence is *cohomological*. whereas the Serre spectral sequence has a *homological* (and a cohomological) version. How to deduce it?2015-06-03

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