Simple examples for the use of spectral sequences - MathOverflow

Simple examples for the use of spectral sequences

2010-05-02T23:39:00Z

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?

Answer by Greg Muller for Simple examples for the use of spectral sequences

2010-05-03T04:19:59Z

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

$R$ in degree $(0,0)$
$R/x$ in degree $(-1,1)$
$R/(Rx+Ry)$ in degree $(-1,2)$
A non-trivial knights-move map (differential on the second page) from $(0,0)$ to $(-1,2)$ which is the natural quotient map.

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.

Answer by Daniel Moskovich for Simple examples for the use of spectral sequences

2010-05-03T06:39:17Z

The best example I can think of is the Lyndon-Hochschild-Serre spectral sequence in group cohomology. See for instance Chapter VII Section 6 of Brown's Cohomology of Groups.
The spectral sequence, for a group extension $1\to H\to G\to Q \to 1$ (I'm using Brown's notation) and for a G-module M, is of the form $$E^2_{pq}=H_p(Q,H_q(H,M))\Rightarrow H_{p+q}(G,M).$$ Let's use this to calculate the third integral homology of the dihedral group D_2n, for $n$ an odd integer. The group extension is $$1\to C_n\to D_{2n}\to C_2 \to 1,$$ and the corresponding Lyndon-Hochschild-Serre spectral sequence is $$E^2_{pq}=H_p(C_2,H_q(C_n,\mathbb{Z}))\Rightarrow H_{3}(D_{2n},\mathbb{Z}),$$ where p and q add up to 3. The integral homology of a cyclic group C_m is $\mathbb{Z}$ for $q=0$, vanishes for $q\in\mathbb{N}^\ast$ even, and is $\mathbb{Z}/m\mathbb{Z}$ for $q$ odd. Plug this information into the Lyndon-Hochschild-Serre spectral sequence, and you find $$H_{3}(D_{2n},\mathbb{Z})=H_0(C_2,\mathbb{Z}/n\mathbb{Z})\oplus H_3(C_2,\mathbb{Z})\simeq \mathbb{Z}/2n\mathbb{Z}.$$ This is easy and elegant calculation in my opinion, and occurs in practice in knot theory ($H_{3}(D_{2n},\mathbb{Z})$ turns out to be isomorphic to the relative bordism group of Fox n-coloured knots).

Answer by Kevin Buzzard for Simple examples for the use of spectral sequences

2010-05-03T08:22:03Z

This is not the most profound answer but it's something that came up last week when I was reading a paper. The author wanted to prove that a certain obstruction to a problem was zero, and the obstruction lived in an $H^2$ that looked a bit scary: it was $H^2(W,V)$ with $W$ a local Weil group and $V$ a finite-dimensional vector space over a field of characteristic zero (I'll tell you all you need to know about this Weil group in a sec, in case you don't know what one is; it's a topological group coming up in number theory). But then I realised the obstruction was really in the image of $H^2(W/C,V)$ with $C$ a compact open subgroup of $W$ (a finite index subgroup of inertia).

But now I'm done because $W/C$ has a two-step filtration with a finite sub and a quotient isomorphic to $\mathbf{Z}$, and I know finite groups have no cohomology in char 0 in degrees 1 or more, and $\mathbf{Z}$ is the fundamental group of a 1-dimensional thing so it has no cohomology in degree 2 or more, and so by Hochschild-Serre, a calculation I can even do in my head in this example, there are no non-zero terms to build $H^2(W/C,V)$ from in $E_2$ and hence in $E_\infty$ so this group vanishes. I can do this calculation without even pulling out a piece of paper.

I'm sure if I were more "group-theoretic" I would have a much clearer picture about what was going on, but Hochschild-Serre just explains to me in a very concrete way how the cohomology of a group is built from the cohomology of its subs and quotients, and is definitely something I carry around in my "useful tools" bag.

Answer by Timo Keller for Simple examples for the use of spectral sequences

2012-01-29T18:51:10Z

Non-trivial spectral sequences occur when calculating the homotopy groups of tmf at 2 or 3: http://www.math.uiuc.edu/~rezk/512-spr2001-notes.pdf §16 ff., see also Tilman Bauer's article http://arxiv.org/abs/math/0311328

Answer by Lalit Jain for Simple examples for the use of spectral sequences

2012-01-29T19:29:48Z

I can't recommend the following document by Tom Weston enough. It introduces spectral sequences rapidly and at a comfortable level of generality. It then applies the Hochschild-Serre sequence to group cohomology.