Patrick Brosnan pbrosnan@gmail.com 12 January 2009 01:38 To: A.Holmstrom@dpmms.cam.ac.uk Hi,
I found your web blog on Harada and I was about to post to it about questions I have about the paper. But I decided to email you instead. (The reason I don’t post is that I keep having moral doubts about whether I should send my comments directly to Harada before posting. But I also hesitate to send my comments to Harada without thoroughly reading the paper and I don’t have time to do that.)
So here goes. I haven’t looked at the second paper but I’ve looked at the first a bit. The main thing that worries me is his claim in 3.2.4 that the K-theory of his semi-simple category is the filtered colimit of K-theory of Artinian algebras over k. I don’t see how to justify this. I wish he would provide a reference.
There is also something that he says in 4.0.2 that is related and simply wrong. There he claims that an Artinian algebra over a field k is isomorphic to a direct sum of Mat(D_i, n_i) where D_i is a division algbra finite over k and n_i an integer. But, certainly the complex numbers are Artinian and semi-simple over the rationals but not of the above form.
It is possible that he has confused “algebra over k which is Artinian” with “Artinian k-module which is an algebra”=“finite dim k-algebra”. Then maybe he should adjust the claim in 3.2.4 to say that the K-theory of his semi-simple category is a filtered colimit of K-theories of finite dimensional algebras. I still do not know how to justify this claim.
Best,
Patrick Andreas Holmstrom andreas.holmstrom@gmail.com 13 January 2009 23:08 To: Patrick Brosnan pbrosnan@gmail.com Hi Patrick,
Thanks for your email, this is very interesting. I emailed Harada about two weeks ago about the issue I wrote about, but he has not replied so far. I agree that maybe it’s a bit unfair to post things publicly without giving him a chance to explain things first, but on the other hand I think it would be very valuable to share thoughts about these preprints, since potentially they contain some really important ideas. If you want to post your observations I think it might be of benefit to others, but do whatever you think is best.
I myself is sort of hesitant about whether to invest a lot of time in this or not. If I do, I will keep posting what I find out, but probably I will work on other things instead. If at some point you look into this again, and you find other mistakes or are able to verify some things, I would of course be very interested.
Best regards,
Andreas
nLab page on Brosnan