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Michi’s blog » archive for June, 2006

Group cohomology revisited: Quaternionic unit group

  • June 20th, 2006
  • 9:54 am

In a previous installment, we calculated H^*(D_8) with some amount of success. For that post, I said that I was going to calculate the cohomologies of D_8 and of U(\mathbb H) by hand – and I’ve been at it for the latter group since then. With some help from my advisor – mainly with executing the obvious algorithms far enough that I get decent material to work with – I know have it.

The group

So, for starters, we need a presentation of \mathbb H such that we can work well with it. We all know that \mathbb H=\langle i,j,k|ij=-k, jk=-i, ki=-j, ijk=i^2=j^2=k^2=-1\rangle. So due to ij=-k and i^2=-1, we can just pick any two of the i,j,k and call them x and y. Then x^4=e, y^2=x^2 and iji=ik=j so xyx=y. This gives us the presentation \mathbb H=\langle x,y|x^4,y^2=x^2,xyx=y\rangle

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Admin: WordPress 2.0

  • June 19th, 2006
  • 3:13 pm

This blog just migrated to WP 2.0. Should anything be odd, please notify me.

The migration was to a large portion motivated by commenting problems I’ve been told about. I hope that my readers out there will be able to comment now; possibly even without logging in!

(48 words)

Weekly Report: Power tourism

  • June 18th, 2006
  • 11:15 am

It’s been a while since I managed to write one of these. The reason is simple enough – my weekends have been packed; and I don’t get around to it during the weeks.

During the last three weekends, first my parents and my brother, and then for the last two and the week inbetween my fiancée, have been visiting me in Jena. Thus, I have covered more ground in these three weeks when it comes to tourism than I probably will be able to do in several months. I have seen the Blue Man Group in Berlin (WOW!), I have seen the Dornburg, the Feengrotten and Weimar. I have eaten at the expensive luxurious restaurant at the top of the old university tower in Jena (it’s bloody scary, but quite cool – the restaurant is on the 29th floor; in a city where only one single house goes above 10 floors).

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More group cohomology

  • June 8th, 2006
  • 10:10 pm

My advisor told me to go hit D_8 and U(\mathbb H) 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 \mathbb F_2 with \mathbb F_2D_8, he walked me through his Shiny! Gr�bner 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 H^{\leq 2}(D_8,\mathbb F_2) and then peeked into Carlson, et.al. for the Big List of 2-group cohomologies to see that all interesting stuff happens in H^{\leq 2} 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!

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First academic publication!

  • June 6th, 2006
  • 3:03 pm

I just received in the mail a bunch of prints. Of my article “Computation of Poincaré-Betti series for monomial rings”, produced from my Master’s thesis for the “School and workshop on computational algebra for algebraic geometry and statistics” in Torino 2004. It is now being published in the Rendiconti di Istituto Matematico di Universita di Trieste, on pages 85-94 of Vol. XXXVII (2005).

Damn, it feels good. Reviewed and everything. If you’re curious, my manuscript can be found at http://math.su.se/~mik/torino.pdf or at the arXiv as math.AC/0502348.

(87 words)