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Michi’s blog » archive for May, 2006

 My first group cohomology – did I screw up?

  • May 21st, 2006
  • 1:42 pm

I thought the seminar on tuesday would possibly benefit from something not very often seen – explicit examples. So I started working through one. I wanted to calculate H^*(C_8,\mathbb F_2) and give explicitly in a series of ways the product structure – as Yoneda splices, as chain map compositions and as cup products.

Now, \mathbb F_2 has a very nice resolution as a \mathbb F_2C_8-module – all cyclic (finite) groups have canonically a really cute minimal resolution – given by

\to \mathbb F_2C_8\to\mathbb F_2C_8\to\mathbb F_2C_8\to\mathbb F_2\to 0

with the last map taking \epsilon\colon a_0+a_1g+a_2g^2+\dots+a_7g^7\mapsto a_0+a_1+\dots+a_7 and the other maps alternatingly being multiplication with \partial_1\colon 1+g+g^2+\dots+g^7 and with \partial_0\colon g-1.

So this gives as a nice projective (in fact: free) resolution to work with. We now can observe that \Hom_{\mathbb F_2C_8}(\mathbb F_2C_8,\mathbb F_2)=\Hom_{\mathbb F_2C_8}(\mathbb F_2,\mathbb F_2)=\mathbb F_2 since any map has to respect the group action, which is trivial on \mathbb F_2, and so any map is determined by its value on 1. Thus we get the sequence of dual modules

0\to\mathbb F_2\to\mathbb F_2\to\mathbb F_2\to\dots

 Weekly Report: From 0 to 100 in 6 seconds flat!

  • May 20th, 2006
  • 2:36 pm

This is the second weekend in a row spent to more or less large part in the office, working with the product structures on cohomology. Reason for this is that I’m getting my share of the department seminars now – I’m to walk us through the Yoneda cohomology product; the cohomology-as-Hom-in-a-derived-category viewpoint; their equivalence to one another as well as to the cup product; and then talk about restriction and corestriction (i.e. what happens to cohomology when we go between kG- and kH-modules for H a subgroup of G)

This is all not really very bad – I really, really, REALLY need to get a solid grip of this myself too. Only; when I started working on it, I thought I had 4 days and not 11 to prepare in – and dove right into it. Maybe a bit too deeply, so when I (monday) found out I didn’t need to get it done THAT quickly, I kinda dropped most of it for the rest of the week. And now, I need to find a decent proof that cup products = Yoneda products. And I just realized that my books don’t really cover it.

 Weekly Report: Mathematics makes my ears bleed

  • May 14th, 2006
  • 1:50 pm

I’m back in Jena now. The last week was spent working myself to the brink of unconsciousness trying to grasp homotopical algebra, simplicial objects, model categories and any and all things Alex sent my way. With some 6-8 hours each day spent on lectures and discussions explicitly held to enable me to understand what was going on, I ended up being halfdead from the mental exercise.

In addition, since I was back in Sweden, outside lecture times was spent almost exclusively socializing in one way or another. Meeting friends. Shopping. Watching movies and spex. And then top it off with an endlessly long trip back.

 Weekly Report: Preparing to leave

  • May 7th, 2006
  • 5:11 pm

I left Jena going to Stockholm on Saturday. Thus, much of the week past has been spent in preparation for the trip – reading up on homotopical algebra; getting all the paperwork together and getting my things together for the trip.

Along with “Make sure you learn homotopical algebra” and “Get back primed and ready to teach when you come”, I also was instructed by my advisor to get in touch with $MATHEMATICIAN, who currently resides at Mittag-Leffler and whom he knew from earlier. He is, I am told, very good, very knowledgeable and definitely a resource to be tapped if I should have any chance of it whatsoever.

I desperately, sincerely need to get a better cheap travel route to Jena. This trip now took me €150-170 somewhere, but had a travel time of more than 13 hours. There has GOT to be a better way to go.

 Weekly Report: Forgot all about it!

  • May 4th, 2006
  • 2:58 pm

Right. It’s thursday. And I had some sort of hopes to do my weekly reports on saturdays. Only, last saturday found me back in Nuremburg, in the middle of a marathon party-after-party session with the RPG crowd there.

Last week was very much characterized by getting various conditions for my being allowed to go to Sweden next week, and getting various bits and pieces of general paperwork in order. In addition to that, I held my first lesson – an examples class in Algebra. Right now, we’re doing modules: general definitions and then the structure theorem for finitely generated modules over a PID. I have already noticed for my self what has been painfully obvious from observing bloggers and friends whining about their students – it’s obvious as soon as you set foot in the classroom who knows what’s going on; and these are the only ones who will give you any sort of life indication. I already started despairing about getting reactions from the people not running up to speed – especially since these also don’t bother handing in any kind of work for me to correct either. End result: I have no idea if I’m doing any good for those who need me, and only get responses from those who don’t. The teacher’s lament.