# Examples spectral sequence

I have to make a talk about spectral sequences, so I'd like to present some concrete examples of computation, after the general definition. I'd like to present three examples of spectral sequences: Cech-De Rham s.s. in order to compute the cohomology groups of complex projective space $\mathbb{C}P^n$, Serre s.s. and the third I'd like to find a s.s. related to homological algebra or algebraic geometry, for example regarding Koszul complex. Can you give me any idea to make some concrete examples of Serre s.s. and Koszul s.s.? (and some bibliography...) Thank you!

Differential Forms in Algebraic Topology by Bott and Tu has many examples that you could present if you want. A User's Guide to Spectral Sequences by McCleary has also tons and tons of examples.

Here are some examples off the top of my head:

• The Serre SS for computing the homology of $\Omega S^n$ ("easy"), of $BU_n$ and $U_n$ ("hard").

• In homological algebra, the Künneth SS or the universal coefficient SS can probably give examples of computations.

• In algebraic geometry, there's the Leray SS, depending on your familiarity with the concepts involved.