My advisor told me to go hit and as my next two cohomology calculation projects; try to do them with resolutions by hand so that I get a feeling for what’s going on. After failing spectacularily both at getting a resolution of with , he walked me through his Shiny! Gröbner base method to get resolutions with free modules over finite p-group algebras. Armed with the minimal resolution, I sat down and started hunting products; and finally found the cohomology ring.
Or … to be exact, I found and then peeked into Carlson, et.al. for the Big List of 2-group cohomologies to see that all interesting stuff happens in anyway.
So for the benefit of any and all readers who want to see what it looks like, I’m going to walk through it again here. Nonono, you don’t need to flee all of you – just skip this entry if it’s that scary!
So. Before starting, we need to be clear about what looks like, and what looks like. So, I will choose a presentation with merits for this particular calculation: . This is a presentation by two mirrorings; and the “usual” presentation of a mirror and a 90-degree rotation can be recovered by the map that sends and .
Now, the algebra, we choose to be presented as with and . Hacking at these relations for a few short moments convinces you that they are equivalent to the group relations.
, and follows rather immediately with the augmentation map, and from sending and to the minimal generators of the augmentation ideal.
Now, the kernel of can be calculated by eliminating the variable from the ideal generated by the group algebra relations and the expressions . This is a (rather) straightforward Gröbner base calculation, and we get at the end the eliminated generators
so we get a kernel with three generators.
By a similar calculation, we get the next kernel . This gives us the first few bits of the minimal free resolution as
So, this calculation gives us a minimal resolution, and thus the presentation as a graded module of our cohomology ring (or at least the portion in degree lower or equal to 3); it remains to be seen what happens with the cohomology product.
So, in degree 0, we have just the constants of the cohomology ring, and nothing very exciting will happen. In degree 1, we have two elements – let’s call them . For their interactions, we want to lift the maps given by ax+by in
Now, obviously will make the beginning commute. Then, makes the square commute, and finally makes the last square commute. Finding these matrices is just a tedium of multiplying corresponding matrices to get the compositions of maps, and then setting the coefficients just so. I won’t enter into more details than that.
Similarily, for degree 2, we get a similar set of matrices, parametrized by a map as and .
How does this help us find the products? Well, for and in the group cohomology, lifting to a chain map – as we have done here for all possible maps in degrees 1 and 2 – and then composing at the corresponding position with a representative for gives a representative for (or possibly the other way around). Now the cohomology in each degree is spanned by just the augmentation on each component; call them for the augmentation of component i in degree j. So, we can calculate, first, all products of things in degree 1:
so we see that , that and that . So we have a relation, and two generators, and in addition two of the three generators of degree 2 already. Let’s call that third generator z, and keep in mind that z lives in degree 2.
This gives us some more products that we can calculate.
We could lift the second degree elements, and with that second lift verify these results, but it’s scarcely necessary. With these calculations, we have the cohomology ring in degrees lower than or equal to 3. Going further means more Gröbner bases, more liftings to chain maps, and I haven’t the energy right now. On the other hand, sneakpeeking in Carlson et.al. reveals that those who have done this -with- a condition to test whether the whole ring is in there – also find three generators, in degrees 1 and 2, and with a single relation.
Thus I have calculated