Talk:Grothendieck group

I quickly turned the equations into TeX, but obviously, this page needs to be improved. Here are some suggestions. The page is supposed to be about the Grothendieck group of an additive category. If it remains as such, the notion of "exact sequence" used in the definition needs to be clarified. Another option (in my opinion better) is to change the subtitle of this page to "of an exact category", and then to consider various exact structures on an additive category. --Baptiste Calmes 21:57, 16 April 2012 (CEST)

I have modified the subtitle as you proposed and also modified the beginning of the article according to Talk:EoM:This_project#Preferred_style. Some MR/ZBL entries and MSC were also added. Remark: Very often, in the references the letter K from "K-theory" is encoded by a png file, which should be removed as well. --Ulf Rehmann 22:59, 16 April 2012 (CEST)

Yes, sorry for missing these. --Baptiste Calmes 15:37, 17 April 2012 (CEST)

Sorry, the 'exact' did not make sense without further change. But the idea should be pursued, I think. --Ulf Rehmann 23:04, 16 April 2012 (CEST)

Yes, this is exactly the problem, and I have no time to make theses changes right away. But as it stands, the page is already slightly strange, because it is unclear what the author meant by "an exact sequence" in an additive category. Does it mean split exact (the middle terme is isomorphic to the direct sum of the extreme ones)? Or does it mean something more subtle, like the first term is a categorical kernel of the second morphism? And so on... That's why I think one should define instead the Grothendieck group of an exact category, in which a short exact sequence is defined by the very structure of exact category. But then we need a page on exact categories... --Baptiste Calmes 15:37, 17 April 2012 (CEST)