I'm looking for basic examples that show the usefulness of spectral sequences even in the simplest case of spectral sequence of a filtered complex.
All I know are certain "extreme cases", where the spectral sequences collapses very early to yield the acyclicity of the given complex or some quasi-isomorphism to another easier complex (balancing tor, for example).
Is there an example of a useful filtration where one really computes something nontrivial also in the higher sheets?
The examples I have in mind come from topology. For example, the calculation of $H_{\ast}(\Omega{\mathbb S}^n;{\mathbb Z})$ is simply beautiful using the Serre spectral sequence, and one needs to pass to the $n$-th sheet until something happens. Another more difficult example would be the computation of the rational cohomology of $K({\mathbb Z},n)$ by induction on $n$ (depending on the parity of $n$, we get a polynomial algebra or an exterior algebra, if I remember correctly).
Are there similar, but purely algebraic examples which could show the usefulness of spectral sequences to those seeing them the first time?
Best Answer
This isn't exactly what you asked, but its a very simple example that (to me) demonstrates some of the necessity of the complexities of spectral sequences. Consider the ring $R=\mathbb{C}[x,y]$, and consider the module $M=(Rx+Ry)\oplus R/x$. Then the double dual spectral sequence converges to the original module: $$ Ext^{-i}_R(Ext^j_R(M,R),R) \Rightarrow M $$ The second page of this spectral sequence has
The spectral sequence collapses on the third page, with $(Rx+Ry)$ in degree $(0,0)$ and $R/x$ in degree $(-1,1)$. One shortcoming of this example is that you get the same module back, split apart into different components; rather than the associated graded of some interesting filtration. If memory serves, there was a way to tinker with this example to give it that property too, but it escapes me at the moment.