Skip to Content »

Michi’s blog » archive for July, 2006

 Carry bits and group cohomology

  • July 27th, 2006
  • 4:05 pm

Got treated today to a really nice workout in group cohomology; most of which is well worth sharing, since seeing it done once gave me a lot of insight.

So, if we pick \mathbb Z/10 and view it as the set 0,1,2,3,4,5,6,7,8,9 and with the group operation given by a*b = a+b % 10, then one standard 2-cocycle is the function
f(a,b) = \begin{cases}1&a+b\geq10\\0&\text{else}\end{cases}
That this actually does form a cocycle would be the same as requiring
f(b,c)-f(a*b,c)+f(a,b*c)-f(a,b)=0
or regrouped
f(a*b,c)+f(a,b)=f(a,b*c)+f(b,c)
which is to say that the number of carry bits generated when adding three digits does not depend on associativity.

This cocycle classifies the group extension
0 \to \mathbb Z/10 \to \mathbb Z/100 \to \mathbb Z/10 \to 0
with the first map taking a+10\mathbb Z\mapsto 10a+100\mathbb Z and the second taking b+100\mathbb Z\mapsto b+10\mathbb Z

Now, this is a nontrivial extension – which is equivalent to it not being a coboundary – by the following calculation:
Suppose f=dg. Then f(a,b)=g(a)+g(b)-g(a*b). So, since f(0,0)=0, we get g(0)-g(0)+g(0)=0, so g(0)=0. For any b≤8, we also get 0=f(1,b)=g(b)-g(b+1)+g(1), so g(b+1)=g(b)+g(1) and thus by induction, g(b)=bg(1) for all 0≤b≤9.
But, now, 1=f(1,9)=g(9)-g(0)+g(1)=10g(1)=0, which is a contradiction.

 Todo and accomplishments

  • July 24th, 2006
  • 2:55 pm

Accomplished:

  • I am done with the coursework for the past semester. Sent off the TeXed up solution sheets to the webmaster today.
  • My pattern observation seems to hold up surprisingly well – there seems to be a theorem to fetch out there somewhere.
  • I have done most of the dishes. Go me!

Todo:

  • FOCUS! I need to learn triangulated categories. Preferably now. I need to stop playing with Haskell and reading up on group cohomology calculations. Preferably now.
  • After triangulated categories, there’s a wealth of things to look into. High priority are spectral sequences, further group cohomology, diamond lemma, path algebra quotients, A, Massey products, return to model categories. Which order these are done might influence the contents of my PhD thesis significantly.
  • I really need to talk to the university sysadmins about the WLan network.

Progress may be found. Just around the corner. I need a donut.

 Weekly Report: Back up again

  • July 23rd, 2006
  • 10:54 am

The weekly reports have been dead for a while. Reason? The blog has been dead for a while.

Hardware woes

The old computer running this website had some problem all of a sudden about 3 weeks ago. These problems appeared as a complete lockdown of the system – no response to anything. So my brother – with me on the other side of a telephone, tried to reboot the box; but couldn’t get it back up online again. He was headed out to a LARP anyway within hours – and so couldn’t really do much more about it.

Right.

End result? I joined forces with a good friend of mine; we split hardware costs for a slick new box – an Asus barebone box with a 64bit processor and a gig of RAM. It received the harddrive and network interface from the old box, and was with that good to go – only .. processor architecture changed; and so for optimal performance, it’d be a nice idea to actually use a new system install that took advantage of the extra available bitwidth.

 Triangulated categories

  • July 17th, 2006
  • 3:47 pm

One predominant tendency in the algebra/category theory camp is to seek out the minimal set of conditions needed to be able to perform a certain technique, and then codifying this into a specific axiomatic system. Thus, you only need to verify the axioms later on in order to get everything else for free.

One such system is the theory of triangulated categories. This pops up in homological algebra; where you like to work with Tor and Ext – both of which turn out to be derived functors, generalizing the tensor product and the homomorphism set respectively. With the construction of the derived category, we can find a category, in which the tensor product in that category is our Tor, and the hom sets is our Ext.